Sine Theorem 1 Lecture Notes I. teaching material analysis
1, the position and function of teaching materials
In junior high school, students have learned the basic relationship between the sides and angles of a triangle; At the same time, in compulsory 4, students also learned trigonometric functions, plane vectors and so on. These provide a solid foundation for students to learn sine theorem. Sine theorem is a generalization of solving right triangle in junior high school, and it is an important formula to reveal the quantitative relationship between sides and angles of triangle. The content of this section is also the basis for students to learn the following knowledge of solving triangles and geometric calculations, and it often involves solving triangles in physics, industrial production, daily life and other disciplines. According to the position and function of the above textbooks, I have determined the following teaching objectives and difficulties.
2. Teaching objectives
(1) Knowledge objective:
① Guide students to discover the content of sine theorem and explore the proof method of sine theorem;
② Simply apply sine theorem to solve triangles, and initially solve some practical problems related to measurement and geometric calculation.
(2) Ability objectives:
① By studying the quantitative relationship between the angles of a right triangle, we find the sine theorem and experience the process of discovering mathematical laws with special to general thinking methods.
② In the process of using sine theorem to solve triangles, gradually cultivate the ability of applying mathematical knowledge to solve practical social problems.
(3) Emotional goal: Stimulate students' learning motivation and curiosity by setting problem situations, and make them actively participate in bilateral exchange activities. By asking questions, thinking and solving problems, we can cultivate students' good psychological quality of self-confidence and self-improvement. Through the teacher's explanation with examples, we can cultivate students' good study habits and scientific learning attitude. 3. Emphasis and difficulty in teaching
Teaching emphasis: the content of sine theorem, the proof and basic application of sine theorem; Teaching difficulties: exploration and proof of sine theorem;
In order to achieve the above goals and break through the above teaching difficulties, I will adopt the following teaching methods and means.
Second, teaching methods and means
1, teaching methods
In the teaching process, teachers are the dominant and students are the main body, creating a harmonious and pleasant teaching environment. According to the content of this class and students' cognitive level, I mainly use inspiration, perceptual experience and multimedia-assisted teaching.
2. Guidance on learning methods
Transfer of learning situation: Students have obtained the preliminary knowledge of the angular relationship of right triangle in junior high school, which is why students psychologically put forward the problem of how to solve the angular relationship of oblique triangle.
Learning method guidance: guide students to master the thinking method of "observation-guess-proof-application", let students learn in problem situations, then make a concrete analysis of examples, and then consolidate them through observation, summary and practice, and gradually deepen their understanding of new knowledge from concrete to abstract.
3. Teaching methods
The use of multimedia to display pictures greatly attracts students' attention, enlivens the classroom atmosphere and mobilizes students' enthusiasm for solving problems. In order to improve classroom efficiency and facilitate students' hands-on practice, I made a piece of exercise paper for the examples and classroom exercises of this class and distributed it to students before class.
Below I will explain how to use the above teaching methods and means to carry out the teaching process.
Third, the teaching process design
Teaching process:
Lead to the topic
Absorb new knowledge
inductive method
Consolidate new knowledge
arrange work
Fourth, summary and analysis:
According to the research of modern educational psychology, the effective teaching of natural concepts is based on students' existing knowledge structure, so I pay attention to the following aspects in teaching design: First, find the "nearest development zone" between students' existing knowledge structure and new natural concepts; Second, guide students to master new concepts through assimilation.
Third, strive to get out of the misunderstanding that "the concept of nature is fleeting and exercises are overwhelming", and urge yourself and students to enter a new world that emphasizes inquiry, communication and process.
I think the design of this course should follow the basic principles of teaching; Pay attention to the development of students' thinking; Carry out the teacher's understanding of this section; Reflect the principle of "combination of learning and application, combination of learning and application". I hope it will play a good role in cultivating students' thinking quality, establishing mathematical thinking and optimizing psychological quality.
Design Intention: My guiding principle of blackboard writing design: simplicity, intuition and emphasis. The blackboard writing teaching in this class focuses on the middle of the blackboard. In order to deepen students' understanding of sine theorem and its application, examples are put in the middle for the whole class to see.
thank you
Lecture notes on sine theorem 2 hello everyone. Today, the topic of my speech is sine theorem. Below I will introduce my teaching design for this course from the following aspects.
I. teaching material analysis
This section of knowledge is the first section of the first chapter of compulsory five, solving triangles, which is closely related to the basic relationship between the sides and angles of triangles and the judgment of triangle congruence learned in junior high school. In daily life and industrial production, we often encounter the problem of solving triangles, and the relationship between triangles and trigonometric functions is often tested in the college entrance examination. So the knowledge of sine theorem and cosine theorem is very important.
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:
Cognitive goal: by creating problem situations, guide students to discover the content of sine theorem, master the content of sine theorem and its proof method, and make students use sine theorem to solve two basic triangle problems.
Ability goal: guide students to summarize sine theorem from special to general through observation, deduction and comparison, cultivate students' innovative consciousness and observation and logical thinking ability, and realize that geometric problems can be transformed into algebraic problems by using vectors as tools and combining numbers and shapes.
Emotional goal: through the communication, cooperation and evaluation between students and teachers and students, create an equal teaching atmosphere for all students, mobilize their initiative and enthusiasm, and stimulate their interest in learning.
Teaching emphasis: the content of sine theorem, the proof of sine theorem and its basic application. Difficulties in teaching: When knowing the diagonals of two sides and one of them to solve a triangle, judge the number of solutions.
Second, teaching methods
According to the characteristics of the content and arrangement of teaching materials, in order to highlight the key points and break through the difficulties more effectively, based on the development of academic students, following the students' cognitive rules, following the guiding ideology of taking teachers as the leading factor, taking students as the main body and taking training as the main line, the inquiry-based classroom teaching mode is adopted, that is, in the teaching process, under the inspiration and guidance of teachers, students' autonomy and cooperation are the premise, and "the discovery of sine theorem" is the basic exploration content.
Third, study law.
Instruct students to master the thinking method of "observation-conjecture-proof-application", adopt various attempts of individuals, groups and groups to solve problems and doubts, and apply what they have learned to the exploration of the nature of arbitrary triangles. Let students study, observe, analogize, think, explore, summarize and try in the problem scenario, which embodies the students' dominant position, enhances their mathematical thinking ability from special to general, forms a scientific attitude of seeking truth from facts and enhances their perseverance in learning.
Fourth, the teaching process
(1) Create a situation (3 minutes)
"Interest is the best teacher". A good beginning of a class means half the battle. This lesson is introduced by a practical problem, "A triangular model of a worker's master is broken, leaving only the part shown on the right, ∠ A = 47, ∠ B = 53, and the length of AB is1m.. I want to repair this part. Stimulate students' enthusiasm for helping others and interest in learning, so as to enter today's learning topic.
(2) Guess-Reasoning-Proof (15min)
Stimulate students' thinking, and start with a special case that they are familiar with (right triangle) to find the sine theorem. Q: Does this conclusion apply to any triangle? Let the students discuss and guess in groups.
In a triangle, the angle satisfies the relationship with the opposite side.
note:
1, emphasizing that the transformation of conjecture into theorem requires strict theoretical proof.
2. Encourage students to prove it by converting the height into a familiar right triangle.
3. Encourage students to think about what knowledge can link length with trigonometric function, and then think about the level of vector analysis, and take the product of quantity as a tool to prove theorem, which embodies the mathematical idea of combining numbers with shapes.
(3) Summary-Application (3 minutes)
1, sine theorem, discuss which kinds of problems about triangles can be solved.
2. Use sine theorem to solve the side length problem of triangular parts introduced in this lesson. Participating in solving practical problems can stimulate students' knowledge and apply it to practical value.
(4) Give an example (8 minutes)
1, for example 1, in △ABC, it is known that A = 32, B = 8 1, 8, A = 42, 9cm, triangular solution,
The example 1 is very simple, and the result is a unique solution. If the sides between two angles of a triangle and the opposite sides of two angles and one of them are known, the triangle can be solved by sine theorem.
2. Example 2. In δ△ABC, it is known that a = 20 cm, b = 28 cm, a = 40, triangle,
It is difficult for students to make it clear that there are two possibilities for finding angles by sine theorem. Ask students to be familiar with the known two sides and one of them.
Solve all kinds of situations in which one side of a triangle is diagonal. Then give the time to the students.
(5) Class exercises (8 minutes)
1. In △ABC, the following conditions are known: triangular solution, (1) A = 45, C = 30, C = 10 cm (2) A = 60, B = 45, C = 20 cm.
2. In △ABC, the following conditions are known: triangular solution, (1) A = 20cm, B = 1 1cm, B = 30 (2) C = 54cm, B = 39cm, C =1.
Students perform on the blackboard, teachers patrol, find problems and answer them in time.
(6) Summary and reflection (3 minutes)
1, which represents the relationship between each side of the triangle and the sine value of the diagonal.
2. Theorem proof starts from right angle, acute angle and obtuse angle respectively, using the idea of classified discussion.
3. I will use vectors as a tool to combine numbers and shapes and turn geometric problems into algebraic problems.
Reflection on the Teaching of verb (abbreviation of verb)
Starting from practical problems, the sine theorem is finally derived through thinking methods such as conjecture, experiment and induction. The outstanding feature of our research is from special to general. In the whole process of exploration, we not only obtained conclusions, but also mastered the general methods of studying problems. Emphasis is placed on research-based learning methods, students' dominant position, and students' enthusiasm is mobilized to make mathematics teaching become the teaching of mathematics activities.
Lecture Notes on Sine Theorem 3 I. The Position and Function of Teaching Materials
This section of knowledge is the first section of the first chapter of compulsory five, solving triangles, which is closely related to the basic relationship between the sides and angles of triangles and the judgment of triangle congruence learned in junior high school. In daily life and industrial production, we often encounter the problem of solving triangles, and the relationship between triangles and trigonometric functions is often tested in the college entrance examination. So the knowledge of sine theorem is very important.
Second, the analysis of learning situation
As a senior one student, students have mastered the basic trigonometric functions, especially in some special triangles, but it is difficult for students to solve the problems of edges and angles of arbitrary triangles.
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.
According to my teaching content, learning analysis and teaching difficulties, I have set the following teaching objectives.
Analysis of teaching objectives:
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 and draw a conclusion through induction.
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, the analysis of teaching rules
Teaching methods: adopt inquiry-based classroom teaching mode, under the guidance of teachers' inspiration, take students' autonomy and cooperation as the premise, take "the discovery of sine theorem" as the basic exploration content, and take real life as the reference object, so that students' thinking can gradually deepen from the problem to the conclusion of conjecture, the exploration of conjecture and the derivation of theorem.
Learning method: Instruct students to master the thinking method of "observation-guess-proof-application", adopt various attempts of individuals, groups and groups to solve problems and doubts, and apply the learned knowledge to the exploration of the nature of arbitrary triangles. Let students study, observe, compare, think, explore and try in the problem situation, and enhance students' mathematical thinking ability from special to general, as well as their perseverance in learning.
Fourth, the teaching process
(A) the creation of situations, cloth suspected exciting interest
"Interest is the best teacher". A good beginning of a class means half the battle. This lesson is introduced by a practical problem, "A triangular model of a worker's master is broken, leaving only the part shown on the right, ∠ A = 47, ∠ B = 53, and the length of AB is1m.. I want to repair this part. Stimulate students' enthusiasm for helping others and interest in learning, so as to enter today's learning topic.
(2) Explore special circumstances and put forward conjectures.
1, stimulate students' thinking, and start with a special case that they are familiar with (right triangle) to find the sine theorem.
2. Does this conclusion apply to any triangle? Instruct students to divide into groups, and use tools such as scale, protractor and calculator to verify the approximate triangle.
3. Let the students sum up the experimental results and guess:
In a triangle, the angle satisfies the relationship with the opposite side.
This builds confidence for the next proof, and makes students' understanding of the conclusion gradually rise from perceptual to rational.
(C) logical reasoning, proving conjecture
1, emphasizing that the transformation of conjecture into theorem requires strict theoretical proof.
2. Encourage students to prove it by converting the height into a familiar right triangle.
3. Encourage students to think about what knowledge can link length with trigonometric function, and then think about the level of vector analysis, and take the product of quantity as a tool to prove theorem, which embodies the mathematical idea of combining numbers with shapes.
4. Think about whether there are other ways to prove the sine theorem, arrange after-class exercises, prompt, make the circumscribed circle of the triangle to construct a right triangle, or prove it by coordinate method.
(D) Summary, simple application
1, let students describe the sine theorem in words, guide students to discover that the theorem has the beauty of symmetry and harmony, and enhance their enjoyment of mathematical beauty.
2, the content of sine theorem, discuss which kinds of problems about triangles can be solved.
3. Use sine theorem to solve the side length problem of triangular parts introduced in this lesson. Participating in solving practical problems can stimulate students' knowledge and apply it to practical value.
(5) Illustrate and consolidate the theorem.
1, for example 1: In △ABC, it is known that A = 32, B=8 1, 8, a=42, 9cm, triangular solution.
The example 1 is very simple, and the result is a unique solution. If the sides between two angles of a triangle and the opposite sides of two angles and one of them are known, the triangle can be solved by sine theorem.
2. Example 2: In △ABC, it is known that a=20cm, b=28cm and A = 40, and the triangle is solved.
It is difficult for students to make it clear that there are two possibilities for finding angles by sine theorem. Students are required to be familiar with all kinds of triangle solutions when they know the diagonal of two sides and one of them. Then give the time to the students.
(6) Classroom exercises to improve and consolidate.
1.△ABC, the following conditions are known to solve triangles.
( 1)A=45，C=30，c= 10cm(2)A=60，B=45，c=20cm
2. In △ ABC, the following conditions are known to solve triangles.
( 1)a=20cm，b= 1 1cm，B=30 (2)c=54cm，b=39cm，C= 1 15
Students perform on the blackboard, teachers patrol, find problems and answer them in time.
(7) Summing up reflections and raising awareness.
Through the above research process, what knowledge and methods have the students mainly learned? What do you think of this?
1, sine is proved by vector.
The theory embodies the mathematical thought of the combination of numbers and shapes.
2. It expresses the relationship between the sides of the triangle and the sine value of the diagonal.
3. Theorem proof starts from right angle, acute angle and obtuse angle respectively, using the idea of classified discussion.
Starting from practical problems, the sine theorem is finally derived through thinking methods such as conjecture, experiment and induction. The outstanding feature of our research is from special to general. In the whole process of exploration, we not only obtained conclusions, but also mastered the general methods of studying problems. Emphasis is placed on research-based learning methods, students' dominant position, and students' enthusiasm is mobilized to make mathematics teaching become the teaching of mathematics activities. )
(eight) task extension, independent inquiry.
If two sides of a triangle and their included angles are known, what about a third side? If sine theorem is not applicable, then naturally transition to the next section, cosine theorem. Assign homework and preview the next section.
Lecture Notes on Sine Theorem IV Dear experts and judges:
Hello everyone!
I'm fwsi, a math teacher at X Middle School in X County. The topic of my speech today is: Sine Theorem, Chapter 1, Section 1, Lesson 1, the required mathematics textbook of version A curriculum standard in ordinary senior high schools. According to the requirements of the new curriculum standard and my understanding of the teaching materials, I will elaborate my design and conception from the following aspects.
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 modeling process of abstracting "practical problems" into "mathematical problems". At the same time, in the process of solving problems, we can feel the power of mathematics, and further cultivate students' interest in learning mathematics and their consciousness 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-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 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")
Cite 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?
Let the students do it by themselves at this time. If they feel that they have difficulty in solving it, they can also study in groups around the table or at the same table, encourage students to prove this conclusion in different ways, and let students with different methods show it on the blackboard during the inspection. 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.
Question 5: According to our research just now, it shows that this conclusion holds true in both right-angled triangles and acute-angled triangles. Then, do we have a bolder guess that the acute angle ⊿ABC in the condition is replaced by the obtuse angle ⊿ABC, and the other things remain unchanged, and this conclusion still holds? You can't just say that it is established, but you must be able to carry out strict theoretical proof. Do you have this ability? Next, I hope you can tell me with your strength. Let's leave now. (Enlighten and guide students to learn and prove in many ways, especially vector method, which is still needed in the proof of cosine theorem in the next section, so we must inspire students to complete the proof with vector method. )
Give students the opportunity and time to practice, let students really participate in the process of solving problems, and let students know and improve their thinking methods and habits in the practice of learning mathematics. At the same time, considering the poor foundation of some students, individuals or groups may not be able to complete the inquiry task. At the same time, the teacher passed the examination and asked the students who had proved the conclusion in advance to finish it on the blackboard. On the one hand, it affirms the advanced nature of the students who finished first, exercises the writing standardization of the students in the process of solving problems on the blackboard, and also gives a reference to the students who can't start, so as not to waste time.
Question 6: Can we draw a general conclusion from this? Can you summarize it in more concise language? Ok, this is the main content of our lesson, the famous sine theorem (the topics on the blackboard are marked with red chalk at this time)
Teacher's explanation: Tell everyone that this famous sine theorem was first discovered and proved by the famous Iranian astronomer Abu Rewi (940-998). Albee Rooney (973- 1048) from Central Asia gave a proof of triangle sine theorem. It is also said that the proof of sine theorem was obtained by Azerbaijanis in13rd century on the basis of systematically sorting out predecessors' achievements. Anyway, we say that people discovered this conclusion full of mathematical beauty before 1000 years ago, which is a miracle in the history of human mathematics. The teacher hopes that in the 2 1 century, you can also develop a so-and-so theorem that future generations admire, and then I will become a mathematician's teacher. Of course, whether the teacher's hope can come true depends on everyone.
Through this explanation, some contents of the history of mathematics are infiltrated, which not only edifies students with the beauty of mathematics, but also stimulates their enthusiasm for learning scientific and cultural knowledge.
(D) to strengthen understanding and simple application
Please read pages 2 to 3 of our textbook to the example 1, and learn the definition of triangle by yourself.
Let students read books and slow down the pace, which is beneficial for students to digest and absorb the content just now. At the same time, teachers can use this time to help individual students with learning difficulties, reduce backward students, and cultivate students' good habit of reading consciously.
After we have studied the sine theorem, what do you think its application is? What problems can he solve in the triangle? Let's start with a simple question:
Question 7: (textbook example1) ⊿ in ABC, A=30 is known? ，B=75？ , a=40cm, triangular solution.
This question is very simple. Let two students finish it on the blackboard, and the other students finish it in the exercise book below. Students can discuss in a low voice, and after the completion, the teacher will make necessary comments according to the problems found in the students' exercises. )
Give students enough time and opportunities to do it by themselves, because this problem is the only solution, and create conditions for students to understand when the triangle has the only solution in the future.
Intensive exercise
Let all the students finish the first exercise on page 4 of the textbook within a limited time, and find two students to the blackboard.
Question 8: (textbook example 2) in ⊿ABC, a=20cm, b=28cm, A=30? Solve triangles.
Example 2 is more difficult, the purpose is to make students clear that there are two possibilities to use sine theorem, and at the same time guide students to learn through the comparative example 1 When does a triangle have a unique solution? Why? Encourage students who have the spare capacity to study to explore and discover the contents of the 8-page textbook: "Re-discussing the triangle"
(5) Summarize and deepen the expansion.
1, sine theorem
2. Proof method of sine theorem
3. Application of Sine Theorem
4. Mathematical ideas and methods involved.
At the same time, teachers and students summarize the harvest of this lesson, guide students to learn to summarize by themselves, and let students further review and experience the process of knowledge formation, development and perfection.
(six) assignments, consolidate and improve
1, textbook 10 page exercise 1, 1 group a1.
2. Students who can learn explore the problem of group B 10 on page 1 and experience other proof methods of sine theorem.
It is proved that if the radius of the circumscribed circle of a triangle is r, then A = 2RSINA, B = 2RSINB and C = 2RSINC.
Designing homework with different gradients for students of different levels and respecting students' personality differences are conducive to implementing the teaching principle of teaching students in accordance with their aptitude.
(7) blackboard design: (omitted)