Lecture Draft of Hyperbola and its Standard Equation 1 I. teaching material analysis
1, current situation of teaching materials
This lesson is the first lesson in the third quarter of the second chapter of the new curriculum version A, and it is an elective course of 2- 1 It is based on students' study of straight lines, circles and ellipses, and also paves the way for parabola and its standard equation.
2, the role of teaching materials (important model, the combination of numbers and shapes)
Conic curve is an important geometric model, which has many geometric properties and is widely used in daily life, production and science and technology. At the same time, conic curve is also an important material that embodies the idea of combining numbers with shapes.
3. Design concept: reflect the requirements of quality education and new curriculum concept, integrate the three-dimensional teaching objectives of "knowledge and skills", "process and method" and "emotional attitude and values", use the school blog platform for online teaching, and highlight the interactivity, thinking, effectiveness and innovation of classroom teaching. Pay attention to the experience of students' learning process, and embody the learning style of autonomy, cooperation and inquiry; Pay attention to the cultivation of mathematical basic ability and mastery of basic knowledge, pay attention to the education of mathematical thinking methods, and reflect the frontier of mathematics and its connection with science, technology and society; In the teaching process, it reflects the role of process evaluation in the development of students and the effective guidance of teachers.
Second, the target analysis
1. Knowledge and skills objectives
① Understand the definition of hyperbola.
② The standard equation of hyperbola can be solved according to known conditions.
③ Further feel the concept of curve equation and understand the basic method of establishing curve equation.
2. Process and method objectives
(1) improve the ability to solve geometric problems with coordinate method and calculation ability.
(2) Cultivate students to study problems with numbers and shapes.
③ Cultivate students' analogical reasoning ability, observation ability, induction ability and exploration and discovery ability.
3. Emotion, attitude and values goals
① Experience the acquisition process of hyperbola and its standard equation, and feel the influence of mathematical beauty.
② Through active exploration, cooperation and communication, feel the pleasure of exploration and successful experience, and appreciate the rationality and rigor of mathematics.
③ Develop a scientific attitude of seeking truth from facts and persistent learning spirit, and form a positive attitude towards learning mathematics knowledge.
4. Key points and difficulties
Based on the above analysis, I will determine the teaching emphases and difficulties of this course as follows:
① Emphasis: Feel the basic process of establishing curve equation, and master the standard equation of hyperbola and its derivation method.
② Difficulties: Derivation of hyperbolic standard equation.
Third, the analysis of learning situation:
1. Knowledge: Students have learned straight lines, circles and ellipses, basically mastered the general method of solving curve equations, simplified equations with two roots, and have some experience in the thinking method of combining numbers with shapes and analogical reasoning.
2. Ability: Students are proficient in basic computer operations, have a certain learning foundation and the ability to analyze and solve problems, and have a certain group communication ability and collaborative discussion learning ability.
Fourthly, the analysis of teaching law.
In teaching methods, inquiry teaching method and heuristic teaching method are mainly adopted. Inquiry learning is to make full use of the characteristics of young students' creativity and curiosity, courage to think and act, and strong interest in new things. Let students analyze, discuss and solve problems consciously and creatively according to the requirements of teaching objectives and the known conditions in the topic.
Heuristic teaching method is based on inspiration and guidance, and adopts the form of questioning to gradually let students carry out inquiry learning. By creating situations, students' existing learning experience can be fully mobilized, so that students can experience the process of "observation-guess-proof-application" and discover new knowledge, and their subconscious curiosity can be transformed into innovative consciousness of consciously seeking knowledge. Through practical operation, the newly generated mathematical knowledge is improved, the students' ability to use their hands and brains is improved, and the comprehensive quality of research and exploration is enhanced.
The new curriculum advocates "autonomy, cooperation and inquiry" learning, guiding students to explore and discover knowledge independently; Support students' active learning activities by designing questions and help them become the main body of learning activities; Create real problem situations and induce them to explore and solve problems. And pay attention to cultivating students' practical ability.
Fifth, talk about the teaching process.
Teaching link
teaching process
Design intent
Introduction to comments
Psychology emphasizes that learning is carried out on the basis of the existing cognitive structure, so that students can use the relevant knowledge and experience in the original cognitive structure to promote the construction of new knowledge independently under the guidance of teachers. This link can not only make students review the old and learn the new, but also pave the way for the later' learning'.
Definition of hyperbola
This paper introduces the definition of hyperbola through the experimental exploration of textbooks (shown in the form of animation): the absolute value of the difference between the distances of two fixed points on a plane is equal to a constant (less than) point set.
Symbolic representation: ()
In which: focus-; Focal length-(set to);
Set constant
Thinking:
1, what is the trajectory of point m after "absolute value" is removed? (Shown in the form of animation)
2. If it is a constant, what is the trajectory of point M? (Shown in the form of animation)
Constructivism theory holds that learning is a process in which students actively construct knowledge. Therefore, students should be allowed to experience the formation and development of knowledge in specific problem situations, abstract practical problems into mathematical models, and explain and apply them. The key to classroom teaching is to stimulate students' thirst for knowledge, so that students can actively participate in and discover learning.
2. By asking questions, students are gradually introduced into the problem situation, and through the interaction between teachers and students, students can learn to think and learn in the problem and finally solve the problem. At the same time, the problem has a certain gradient, which has a certain guiding and enlightening effect on students' thinking.
Standard hyperbolic equation
1. Review the general steps of finding curve equation: building system, setting points, formulating, simplifying and testing.
2. Derive the standard equation of hyperbola focusing on X axis and Y axis.
Students are divided into two groups. One group deduces the standard equation of hyperbola with focus on X axis, the other group deduces the standard equation of hyperbola with focus on Y axis, and finally exchange conclusions.
3. Compare two standard equations.
Two explanations:
① Relationship:
(2) How to judge the position of the focus: Look at the positive and negative coefficients in front and see which one is positive on the corresponding axis. (Formula: Pay attention to pros and cons! )
1. After comparing how to simplify the equation, I choose to let students simplify, let them experience the hardships of simplifying the equation, experience exercise, try to succeed, and improve their enthusiasm for participating in the teaching process.
2. After getting the hyperbolic standard equation, my students and I summarized the steps of deriving the hyperbolic standard equation, with the aim of further strengthening the general steps of finding the curve equation and letting students enjoy the joy of success.
3. Embody the idea of analogical reasoning and cultivate students' ability of induction and analogical reasoning.
4. In the process of derivation, I ordered that one is to beautify the equation and make it symmetrical, and the other is to pave the way for the later study of geometric properties.
Case analysis
The teaching of example 1 is to make students clear that if the focal coordinates of hyperbola (or in an equation) are required, the equation must be transformed into a standard equation.
Through example 2, students can understand that the standard equation of hyperbola mainly determines two elements: first, the position of hyperbola is determined by the focus; The second is the shape of hyperbola, which is determined by its origin.
Example 3 is the practical application of hyperbola. The key is to use the definition of hyperbola to solve problems and pay attention to the position of focus.
Course summary
In order to let students build their own knowledge system, I ask students to summarize what they have learned. I think this can not only cultivate students' generalization ability, but also create a democratic and harmonious relationship between teachers and students.
Online testing
Through the network platform of the school, students can consolidate the basic knowledge in time and understand the answers of the whole class. The teacher makes comments.
Keep abreast of students' mastery.
work arrangement
Hand in: People's Education Edition Senior High School Mathematics Elective Course 2- 1
P6 1 Exercise 2.3 A Group 2, 5 B Group 2.
Disjoint: 2.3. 1 hyperbola of the second kind and its standard equation.
Further consolidate the knowledge we have learned in this class.
Six, blackboard design:
First, the definition of hyperbola
Second, the standard equation of hyperbola
1, pay attention to X axis 2, pay attention to Y axis.
Third, the case analysis
Example 1
Example 2
Example 3
I chose this blackboard design in order to make students clearly understand the important content of this lesson.
Seven. Evaluation design
The biggest feature of this lesson is:
(1) We can make full use of network resources in class. For example, we can draw ellipses with geometry sketchpad and flash, so that students can operate and feel the process of things happening. Many rich and interesting learning activities make students truly become the masters of learning.
(2) In the teaching process, I ask questions in gradient. Let all students take the initiative to participate in the whole process of discussion. The questions were put forward one by one, and the students followed my guidance step by step to draw the final conclusion, which fully mobilized the students' enthusiasm for learning.
(3) Check students' mastery of this lesson through online test. After getting the feedback of learning situation, solve it in time and achieve good results.
As a teacher, I always keep in mind in classroom teaching: students are the main body of learning and students are the main body of the classroom; Teachers are only organizers, guides and collaborators of classroom teaching activities. Therefore, when guiding students to get the definition of hyperbola from experimental inquiry and the standard equation of hyperbola from analogy ellipse, I always put myself in the position of organizer, guide and collaborator in the process of explaining examples, so that students can learn independently through activities such as practice, inquiry, induction, analysis and summary, and cultivate their ability to read pictures and summarize.
This course adopts the teaching method of "Task-based Teaching Mode of Mathematics in Network Environment", which enables students to study independently, cooperatively and exploringly. The teaching objectives are clear, the key points are prominent, the difficulties are broken through, the teaching capacity is large, and the classroom teaching design is reasonable. In the teaching process, it can stimulate students' curiosity, pay attention to cultivating students' hands-on ability, guide students to study and learn actively, and use online tests to talk and practice, so that students can get timely.
Lecture notes on hyperbola and its standard equation II. teaching material analysis and its treatment
1, the position and function of teaching materials
Students' initial understanding of conic curve begins with ellipse, and the study of hyperbola is to further deepen and improve its research content. If hyperbola is thoroughly studied and clearly studied, then the study of parabola is logical. So the role of this lesson is to undertake the study of ellipse definition and standard equation vertically, and lay the foundation for the study of simple properties of hyperbola horizontally.
2. Analysis of students' situation:
Before learning this lesson, students have mastered the definition of ellipse and standard equation, and also tried the inquiry learning method, so they have their own basis for exploring and deducing equations in knowledge and learning methods. In addition, senior two students are active in thinking and dare to express themselves. They don't like passively accepting other people's ready-made views, but at the same time they lack the awareness of finding and asking questions.
According to the above analysis of teaching materials and students, and taking into account students' existing cognitive laws, I hope students can achieve the following three teaching objectives.
3. Teaching objectives
(1) Knowledge and skills: Understand the definition of hyperbola and independently derive standard equations;
(2) Process and method: Through the excavation and exploration of definition and standard equation, students can further experience the application of thinking methods such as analogy and combination of numbers and shapes, and improve their observation and exploration ability;
(3) Emotional attitude and values: through the students' exchange and exploration activities under the guidance of teachers, students' interest in learning is stimulated and students are trained to understand the problem from the perspective of contact.
4. Teaching emphases and difficulties
According to the teaching objectives and students' cognitive laws, it is determined that the focus of this course is to understand and master the definition of hyperbola and its standard equation. The difficulty is the derivation of hyperbolic standard equation.
5, teaching material processing:
I have made some adjustments to the teaching content: the textbook is a hyperbolic figure drawn with a string, and I use a geometric sketchpad instead of drawing hyperbolic figures. Because in contrast, the geometric sketchpad is more intuitive. Through the geometry sketchpad, students can not only see the process of hyperbola formation, but also easily see the relationship and difference between ellipse and hyperbola formation.
Second, teaching methods and teaching means
1, teaching methods
The famous mathematician Paulia said, "The best way to learn anything is to discover it yourself."
The definition and standard equation of hyperbola are very similar to those of ellipse, and students have already had some experience in learning ellipse, so I adopted the teaching method of "heuristic inquiry" in this class, focusing on the following two points:
(1) Take analogical thinking as the main line of teaching.
(2) Independent inquiry is a way of learning for students.
2. Teaching methods
Multimedia-assisted teaching is adopted. Reflected in drawing hyperbola with geometric sketchpad. However, we don't simply use animation to show students, but use animation to inspire and guide students to think and arouse their enthusiasm for learning.
Third, the teaching process and design
In order to achieve the teaching goal of this class, better highlight the key points and disperse the difficulties, I divide the teaching process into four stages.
Knowledge introduction, knowledge review, observation animation, and general definition.
At the beginning of the course, I set the following questions to help students review their knowledge:
What is the first definition of (1) ellipse? What are the key words in the definition?
(2) What is the standard equation of ellipse?