Lecture Notes on Pythagorean Theorem 1 I. teaching material analysis:
(A) the status and role of teaching materials
Pythagorean theorem reveals the quantitative relationship among three sides of right triangle from the perspective of knowledge structure, which provides an important theoretical basis for subsequent learning to solve right triangle and has a wide range of applications in real life.
From the perspective of students' cognitive structure, the characteristics of shape are transformed into quantitative relations, which builds a bridge between geometry and algebra. Pythagorean theorem is a good material for patriotism education for students, so it has a very important position and role.
According to the new curriculum standards of mathematics and the cognitive level of eighth grade students, I have determined the following learning goals: knowledge and skills, mathematical thinking, problem solving and emotional attitude. Among them, emotional attitude, with China's mathematical culture as the main line, inspires students to love the long-standing culture of the motherland.
(2) Key points and difficulties
In order to change passive acceptance into active exploration, I have determined that the focus of this lesson is: the exploration process of Pythagorean theorem. Limited to the thinking level of eighth-grade students, I will make Pythagorean theorem discovered by area method (puzzle method) the difficulty of this class, and I will guide students to do experiments to highlight key points and break through difficulties through cooperation and communication.
Second, the analysis of teaching methods
Teaching method Ye Shengtao said, "The teacher's teaching is not the total prize, but the camera induction." Therefore, teachers use geometry to ask questions intuitively, guide students to explore from the shallow to the deep, design experiments for students to verify and understand the thinking methods contained in them.
In order to return the initiative of learning to students, teachers encourage students to adopt hands-on practice, independent exploration and cooperative communication, so that students can personally perceive and experience the formation process of knowledge.
Third, the teaching process
China's mathematical culture has a long history and is profound. In order to let students feel the charm of its inheritance, I design this lesson as the following five links.
First of all, the situation introduces the ancient rhyme and the present style.
Give a set of seven clever eight diagrams, and let the students cooperate with two sets of puzzles. Let the students observe and think about the relationship between the areas of three squares. What kind of triangle do they form, what kind of mathematical mystery do they contain? Education through fun can stimulate students' curiosity and desire to explore.
The second step is to trace the history and decrypt the truth.
The exploration process of Pythagorean theorem is the focus of this lesson. According to the principle of gradual and spiral increase of mathematical knowledge, I designed the following three activities.
Starting from the above problems, the starting point is low, which is conducive to students' participation in exploration. Students can easily find that the isosceles triangle has the following relations. The relationship between areas is skillfully transformed into the relationship between side lengths, which embodies the idea of transformation. It is found that although intuitive, the area calculation is more convincing. The figure is transformed into a figure with edges on the grid line, so as to calculate the figure area, which embodies the idea of combining numbers with shapes. Students will think of the method of "counting grids". Although this method is simple and feasible, it is not suitable for the next step to explore the general right triangle and has limitations. Therefore, teachers should guide students to find the area of square C by "cutting" and "filling", so as to pave the way for exploring the area of complex graphics in the next step.
Break through the shackles of isosceles right-angled triangles and explore whether this conclusion also exists in right-angled triangles. It embodies the cognitive law of "from special to general". The teacher gave right-angled triangles with unit lengths of 3, 4 and 5 respectively, which avoided the mistakes made by students due to inaccurate drawing and laid the foundation for the following proposal of "hook three strands, four strings and five". With the foreshadowing of the previous link, the difficulties are effectively dispersed. When calculating the area of square C, students will show the methods of "cutting" and "filling". Some students may find ways to translate and rotate. Teachers should praise these two new methods, affirm students' research results, and cultivate students' analogy ability, transfer ability and problem exploration ability.
Using the dynamic demonstration of geometry sketchpad, the relationship between geometry and algebra is visualized. When it is a right triangle, the relationship among the three sides remains unchanged by changing the length of the three sides, and when ∠ α is an acute angle or an obtuse angle, the relationship among the three sides changes, further emphasizing that the premise of the proposition must be a right triangle. It deepens students' understanding of Pythagorean theorem and broadens their horizons.
The above three steps are step by step, and students get the proposition 1, thus cultivating students' reasonable reasoning ability and language expression ability.
Perceptual knowledge is not necessarily correct, and reasoning verification confirms our conjecture.
The third step is to bring forth the new and borrow the old to innovate.
The proof method of "Zhao Shuang's String Diagram" given directly in the textbook is a kind of imprisonment for students' thinking. Teachers creatively use teaching materials, liberate students' brains with jigsaw puzzles, and let students use their intelligence to prove Pythagorean theorem. This is the difficulty and focus of teaching. Teachers should give students enough time and space to explore independently, so that students' thinking can collide in mutual discussion and improve in mutual learning. Teachers go deep into students, observe students' inquiry methods, accept students' questions and affirm different solutions to puzzles. Thus, it embodies the teaching concept of "students are the main body of learning, and teachers are organizers, guides and collaborators". Students will find two proof schemes.
Scheme 1 is Zhao Shuang's string diagram. Students explain the demonstration process and reproduce the exploration methods of ancient mathematicians. Scheme 2 is the result of students' own exploration, and the skillful demonstration is similar to scheme 1. The whole exploration process allows students to experience the excavation process from surface to essence, from perceptual reasoning to deductive reasoning, and to appreciate the rigor of mathematics. Through the comparison between "ancient" and "modern" proofs, students can feel the joy of "blowing yellow sand to win gold" and the pride of "shine on you is better than blue". Write Pythagorean Theorem on the blackboard, and then express it in letters to cultivate students' symbol consciousness.
The teacher introduced the meaning of "Gou, Gu and Xian" and the research on Pythagorean Theorem in ancient and modern China and abroad, so that students can feel the mathematical culture and cultivate national pride and patriotism. Use Pythagoras tree to demonstrate dynamically, so that students can realize the exquisiteness and beauty of mathematics.
The fourth step is to take its essence and make the past serve the present.
I designed the following three groups of exercises according to the gradient of "understanding-mastering-using".
(1) correspond to the difficulties and consolidate the knowledge learned.
(2) Examine key points and deepen new knowledge.
(3) Solve problems and feel the application.
The fifth step is to review and reflect on the task extension.
At the end of the class, I encourage students to summarize this class from the requirements of "Four Basics". Then summed up a theorem, two schemes, three ideas and four experiences.
Then assign homework. Hierarchical homework embodies the idea of education for all students.
Lecture Notes on Pythagorean Theorem 21. teaching material analysis
(A) the status of teaching materials
This lesson is the first lesson of Exploring Pythagorean Theorem, a junior high school textbook for nine-year compulsory education. Pythagorean theorem is one of several important theorems in geometry, which reveals the quantitative relationship between three sides in a right triangle. It has played an important role in the development of mathematics and has a wide range of functions in the world at present. Through the study of Pythagorean theorem, students can have a further understanding and understanding of right triangle on the original basis.
(B) Teaching objectives
Knowledge and ability: master Pythagorean theorem and use Pythagorean theorem to solve some simple practical problems.
Process and method: experience the process of exploring and verifying Pythagorean theorem, understand the method of verifying Pythagorean theorem with puzzles, cultivate students' reasonable reasoning consciousness, the habit of active exploration, and feel the combination of numbers and shapes and the thought from special to general.
Emotional attitude and values: inspire students' patriotic enthusiasm, let students experience their sense of accomplishment in trying to draw conclusions, experience mathematics full of exploration and creation, and experience the beauty of mathematics, so as to understand and like mathematics.
(3) Teaching emphasis:
Experience the process of exploring and verifying Pythagorean theorem, and use it to solve some simple practical problems.
Teaching difficulty: using area method (puzzle method) to discover Pythagorean theorem.
The way to highlight key points and break through difficulties: give full play to students' main role, and let students explore in the experiment, understand in the exploration and comprehend in the comprehension through students' hands-on experiments.
Second, the analysis of teaching methods and learning methods:
Analysis of learning situation: Grade eight students have already possessed certain abilities of observation, induction, conjecture and reasoning. They have learned some methods to calculate the area of geometric figures (including cutting and splicing) in primary school, but their awareness and ability to solve problems by using area method and cutting and splicing ideas are not enough. In addition, students generally study and participate in classroom activities more actively, but their cooperation and communication skills need to be strengthened.
Analysis of teaching methods: Based on the characteristics of eighth-grade students and the teaching materials in this section, the teaching adopts the mode of "problem situation-establishing model-explaining application-expanding and consolidating" and chooses the guided inquiry method. Turn the teaching process into a process of students' personal observation, bold guess, independent exploration, cooperation and exchange, and induction.
Analysis of learning methods: under the guidance of teachers' organization, students adopt the discussion learning mode of independent inquiry and cooperative communication, so that students can truly become the masters of learning.
Third, the teaching process design
1. Create situations and ask questions.
2. Experimental operation and model construction
3. Return to life and apply new knowledge.
4. Expansion, consolidation and deepening of knowledge. Feel the harvest and assign homework
(A) the creation of questioning situations
Firefighters rushed to put out the fire on the third floor of the building and learned that each floor was 3 meters high. The fireman is holding a 6.5-meter ladder. If the distance from the bottom of the ladder to the wall base is 2.5 meters, can firefighters enter the third floor to put out the fire?
Design intention: Introducing new courses from practical problems reflects that mathematics comes from real life and people's needs, and also reflects the process of knowledge. The process of solving problems is also a "mathematical" process, which leads to the following links.
Construction of experimental operation model
1, isosceles right triangle (several squares)
2, general right triangle (section)
Question 1: What is the relationship between the areas of squares I, II and III for isosceles right triangle?
Design intention: This will help students to participate in exploration, cultivate their language expression ability and experience the idea of combining numbers with shapes.
Question 2: Do the areas of squares I, II and III have the same relationship for a general right triangle? (Digging and filling method is the difficulty of this section, organize students to cooperate and exchange. )
Design intention: It is not only conducive to breaking through difficulties, but also lays a foundation for inductive conclusion, so that students' ability to analyze and solve problems can be improved invisibly.
Through the above experiments, the Pythagorean theorem is summarized.
Design Intention: Through cooperation and communication, students summed up the rudiment of Pythagorean Theorem, cultivated students' ability of abstract generalization, and at the same time played the main role of students and experienced the cognitive law from special to general.
Return to the new knowledge of life application
Let students solve problems in the opening scene, call for help before responding, enhance students' awareness of learning and using mathematics, and increase the fun and confidence of applying what they have learned.
Fourth, the expansion, consolidation and deepening of knowledge
Basic questions, situational questions and inquiry questions.
Design intention: give a group of questions, divide them into three gradients, practice from shallow to deep, take care of students' individual differences and pay attention to students' personality development. The application of knowledge has been sublimated.
Basic question: The right side of a right triangle is 3, the hypotenuse is 5, and the other right side is X. How many mathematical questions can you ask according to the conditions? Can you solve the problem raised?
Design intention: This question is based on double bases. Students can create their own situations and exercise divergent thinking.
Situation: Xiaoming's mother bought a 29-inch (74 cm) TV set. Xiao Ming measured the TV screen and found that the screen was only 58 cm long and 46 cm wide. He thinks that the salesman must have made a mistake. Do you agree with his idea?
Design intention: to increase students' common sense of life, and also to show that mathematics comes from life and is used in life.
Question: Can a wooden box with a length of 50 cm, a width of 40 cm and a height of 30 cm be put in? Why? Try to explain what you learned today.
Design intention: It is relatively difficult to explore the problem, but teachers use the teaching mode to cooperate and communicate with students, expand students' thinking and develop their spatial imagination.
Five, feel the harvest operation:
What have you gained from this course?
1, textbook exercise 2. 1
2. Collect information about the proof of Pythagorean theorem.
Design description:
1. Explore the theorem by using the area method to create a harmonious and relaxed situation for students and let them experience the combination of numbers and shapes, from special to general thinking methods.
2. Let all students participate and pay attention to the evaluation of student activities. First, the degree of students' participation in activities; The second is the level of thinking and expression of students in activities.
The Third Draft of Pythagorean Theorem Dear judges and teachers,
Hello everyone!
I said that the topic of the class is Pythagorean Theorem in the first lesson of the first section of Chapter 14 of Grade 8 of China Normal University Edition.
Teaching material analysis:
If mathematical thought is a classic old song to solve mathematical problems, then the thought from special to general, mathematical modeling and transformation contained in this lesson are the most active notes in the song! The content of this section is the teaching after learning the quadratic root, which is a follow-up study on the basis that students have mastered the related properties of right triangle, one of several important theorems in middle school mathematics. It reveals the quantitative relationship among the three sides of a right triangle, is one of the main bases for solving a right triangle, is the soul for solving the knowledge of quadrangles and circles, and is widely used in real life.
The discovery, verification and application of Pythagorean Theorem contains rich cultural values and occupies an important position in theory, so this section plays a bridge role in the teaching materials.
Mathematics teaching under the new curriculum standard is not only the imparting of knowledge, but also the cultivation of ability and emotional education. Therefore, according to the position and role of this section in teaching, combined with the characteristics of junior two students who don't like performances and are quiet and restless, I have determined the teaching objectives of this section as follows:
1. Explore and prove Pythagorean theorem by puzzles.
2. Using Pythagorean Theorem to solve simple mathematical problems.
3. Feel the mathematical culture, and realize the diversity of problem-solving methods and the idea of combining numbers with shapes.
According to the requirements of curriculum standards, on the basis of in-depth understanding of teaching materials, I have determined the teaching emphases, difficulties and emphases of this section as follows:
The proof and simple application of Pythagorean theorem is the focus of this section. It is difficult to prove Pythagorean theorem by using jigsaw puzzles, and the key to solve the difficulty is to make full use of various representations of graphic area to construct identities.
In order to clarify the key points, break through the difficulties, grasp the key points, and enable students to achieve the predetermined goals, I analyze the teaching methods and learning methods as follows:
Analysis of teaching methods:
The new curriculum standard emphasizes that students' learning enthusiasm should be stimulated to the maximum extent from their existing experience, and mathematics teachers under the new curriculum should be organizers, guides and collaborators of students' learning activities. Therefore, in view of the focus of the textbook and the cognitive level of junior two students, I take students' full preview as the premise and students' hands-on operation and explanation as the center, so that students can experience the process of doing mathematics, stimulate students' interest in exploration and make the classroom active and interesting. By combining observation, induction, guiding discovery, discussion and other teaching methods, students can fully display their preview results and experience the happiness of success, laying a solid foundation for lifelong learning and development. In order to increase the classroom capacity, create an efficient mathematics classroom for students, and provide students with enough time to engage in mathematics activities, multimedia-assisted teaching is used in the form of tutorial classes.
Analysis of learning methods:
Learning method is a magic weapon for students to regenerate knowledge. In order to solve the problem that students' learning process is a process of understanding things, I first guide students to operate first, and then cooperate and communicate, so as to cultivate students' good learning quality and ability to cooperate with others. Next, I asked students to think independently and urged them to try boldly with special to general ideas, which naturally highlighted the exploration of Pythagorean theorem. Then, through the students' display results, I ask the students to master the key to spelling out graphs in different ways, so as to express the area of graphs in different ways and establish identities. I use my own jigsaw operation and explanation to show the preview results of the difficulty of proving the breakthrough theorem, guide students to write rigorously and reasonably, and cultivate students' logical thinking ability and language expression ability.
In order to fully mobilize students' learning enthusiasm and create an optimized and efficient mathematics classroom, I designed a step-by-step teaching process in the form of a tutorial.
On the premise that students must read 48-52 pages of textbooks and choose 55-56 pages of textbooks for pre-class preparation, * * * is divided into four links for teaching.
1. Exploration of Pythagorean Theorem: Let students preview before class by measuring, calculating and thinking from special to general mathematical ideas, and then pave the way for the exploration of new knowledge by checking the preview results.
2. Proof of Pythagorean Theorem: The proof of Pythagorean Theorem is completed in the form of students' puzzles and explanations of preview results.
3. Application of Pythagorean Theorem: The flexible application of Pythagorean Theorem is realized in the form of classroom practice, students' personality supplement and teachers' appropriate personality supplement.
4. Reflection after learning: in the form of students' summary, guide students to consolidate and sublimate the content of this section from two aspects: knowledge and emotion.
Say something original:
In order to create a harmonious, democratic, equal and efficient mathematics classroom for students, I take the basic concept and overall goal of the new curriculum as the guiding ideology, face all students, choose appropriate breakthrough points and methods, and give full play to the principle of the unity of students' dominant position and teachers' leading role. Pay attention to the cultivation of students' hands-on operation ability in teaching, simplify the complex and turn the abstract into intuition. For example, I take the preview results as the main line, and use students' hands-on operation and explanation instead of time-consuming and laborious drawing, cutting and commenting, which not only allows every student to actively participate, but also cultivates students' language expression ability and logical reasoning ability, and achieves intuitive and efficient results.
In teaching, I pay attention to the creation of humanistic environment, so that the mathematics classroom is full of cordial and democratic atmosphere. For example, I pay attention to students' operation, display, explanation and personality supplement in the whole class, which narrows the distance between mathematics and students and stimulates students' interest in learning. In order to let different students get different development and let everyone learn valuable mathematics, I creatively use teaching materials in teaching, and create a greenhouse project around me as a situation to reflect mathematics life without changing the original intention of the examples. In the form of a changeable question and the adaptation of the senior high school entrance examination questions, the practice is deepened layer by layer, which embodies the beauty of mathematical changes.
Promote a new generation of classrooms in the form of students' personality supplement, cultivate students' innovative thinking to the maximum extent, and let different people have different development in mathematics. This course not only makes the course open, but also provides a space for giving full play to students' wisdom and creative thinking, and creates a composition-style mathematics classroom with unique teaching style. The introduction of multimedia teaching provides students with broad thinking space and time; At the same time, it pays attention to cultivating students' mathematical culture and infiltrating mathematical thoughts, and pays attention to the unity of aesthetic education, moral education and education, such as the message of hope from Pythagoras tree to wisdom tree when summing up.
Lecture Notes of Pythagorean Theorem 4 I. teaching material analysis:
Pythagorean theorem is a very important property of right triangle and one of the most important theorems in geometry. It reveals the quantitative relationship between the three sides of a triangle, which can solve the calculation problem in a right triangle and is one of the main basis for solving a right triangle. Very useful in real life.
When compiling teaching materials, we should pay attention to cultivating students' hands-on operation ability and problem analysis ability, and make students get a more intuitive impression through practical analysis, puzzles and other activities; Understanding Pythagorean Theorem through contact and comparison is beneficial to correct application.
Therefore, the teaching objectives are as follows:
1. Understand and master Pythagorean theorem and its proof.
2. Be able to use Pythagorean theorem and its calculation flexibly.
3. Cultivate students' abilities of observation, comparison, analysis and reasoning.
4. By introducing the achievements of ancient Pythagoras characters in China, we can inspire students' thoughts and feelings of loving the motherland and its long culture, and cultivate their national pride and research spirit.
Second, the teaching focus:
Proof and application of pythagorean theorem.
Third, the teaching difficulties:
Proof of pythagorean theorem.
Fourth, teaching methods and learning methods:
Teaching methods and learning methods are embodied in the whole teaching process. The teaching methods and learning methods of this course reflect the following characteristics:
Give priority to self-study counseling, give full play to the leading role of teachers, stimulate students' desire and interest in learning by various means, organize student activities, and let students actively participate in the whole learning process.
Effectively reflect students' dominant position, let students understand theorems through observation, analysis, discussion, operation and induction, improve their hands-on operation ability, and their ability to analyze and solve problems.
By demonstrating objects, students are guided to observe, operate, analyze and prove, so that students can gain a sense of success in acquiring new knowledge, thus stimulating their desire to learn new knowledge.
Verb (abbreviation of verb) teaching program
The teaching of this section is mainly reflected in students' hands-on and brains. According to students' cognitive rules and learning psychology, the teaching plan is designed as follows:
(A) to create a new situation
1, the story is introduced. More than 3,000 years ago, a man named Shang Gao told the Duke of Zhou that if you fold a ruler into a right angle and connect the two ends, you will form a right triangle. If the hook is 3 and the rope is 4, then the rope is equal to 5. This has aroused students' interest in learning and stimulated their thirst for knowledge.
2. Do all right triangles have this property? Teachers should be good at arousing doubts and let students enter a state of being willing to learn.
3. Write it on the blackboard to show the learning objectives.
(B) the initial perception and understanding of teaching materials
Teachers guide students to learn new knowledge through self-study, which embodies students' awareness of autonomous learning, exercises students' initiative to explore knowledge, and forms good self-study habits.
(3) Ask questions to solve problems and discuss and summarize:
1. Teachers question or students question. How to prove Pythagorean theorem? Through self-study, students above the intermediate level can basically master it, which can stimulate students' desire to express themselves.
2. Teachers guide students to do puzzles and observe and analyze them as required;
(1) What are the characteristics of these two graphs?
(2) Can you write down the areas of these two figures?
(3) How to use Pythagorean theorem? Are there any other forms?
At this time, the teacher organizes students to discuss in groups, arouses the enthusiasm of all students, achieves the effect of everyone's participation, and then communicates with the whole class. First of all, one group of representatives spoke and expounded their understanding of the problem, while the other groups made comments and supplements. Teachers give enlightening guidance in time, and finally teachers and students sum up each other, form a consensus and finally solve the problem.
(4) Consolidate practice and strengthen improvement.
1, show exercises, students answer in groups, and students summarize the law of solving problems. Combine static and dynamic in classroom teaching to avoid causing students fatigue.
2. Give an example of 1. Students try to solve the problem, and teachers and students evaluate it together, so as to deepen the understanding and application of the example. In order to further improve students' ability to use knowledge, we can take the form of mutual evaluation and discussion on the problems in practice, and teachers can take the form of classroom discussion to solve the representative problems in mutual evaluation and discussion, thus highlighting the teaching focus.
(5) Summarize practical feedback.
Guide students to summarize the main points of knowledge and sort out their learning ideas. Distribute self-feedback exercises, and students can complete them independently.
This course aims to create a pleasant and harmonious learning atmosphere, optimize teaching methods, improve classroom teaching efficiency with the help of multimedia, and establish an equal, democratic and harmonious relationship between teachers and students. Strengthen the cooperation between teachers and students, create a classroom atmosphere in which students dare to think, feel and ask questions, make all students lively in teaching activities, and cultivate their innovative spirit and practical ability in learning.