Teaching theme discovery ellipse
Applicable object □ Country 1 □ Country 2 □ Country 3 □ Middle School □ Senior 1 ■ Senior 2 □ Senior 3 □ Other (please specify)
Design concept 1. The idea of ellipse teaching in traditional textbooks is to draw an ellipse first and then teach the definition of ellipse. Although simple and clear, the oval figure will be presented to the students with the movement of the teacher's pen tip. At this time, students who are used to obedience will naturally not ask, "How did you find this rule of ellipse?" Mathematics teachers must reflect: in ellipse teaching, should the definition of ellipse be presented to students as a conclusion, or should it be regarded as an amazing discovery in the process of students exploring knowledge? 2. With the improvement of computer hardware functions and mathematical software, teachers should not be limited to traditional teaching methods, but should make full use of the 2D and 3D functions of computers and the auxiliary teaching provided by the Internet. With the support of mathematical software such as cabri and gsp, teachers lead students to design and operate by themselves, so that the ellipse in students' minds can really move intuitively in front of their eyes. 3. Math software provides students with hands-on opportunities in the learning process, provides an intuitive observation environment for geometry, and allows students to choose their own way to do math, with both frustrated lessons and successful experiences. It improves the previous practice of teachers demonstrating with teaching AIDS and defining students in a pre-arranged order. 4. This teaching plan focuses on the process of students' learning mathematics, and manipulates motion graphics to make ellipse teaching a "student-centered" "doing mathematics" process. What teachers should do is mainly to guide students to think from a different angle, and then expect new and exciting discoveries. The role of teachers is precisely to create inner experience situations for students, create opportunities for dialogue between people and computers, and create the effect of interaction between teachers and students or students, so that students can use the relevant knowledge and experience in the original cognitive structure to assimilate the new knowledge they have learned at present and give new knowledge some meaning.
Teaching objectives 1. Teachers should first learn the basic operations of cabri, cosmo player and other software in the web interface. 2. Although the content of teaching is the definition and nature of ellipse, with the help of the new technology of mathematical software, teachers may not all use traditional books to teach, but interact with students through the interface of mathematical software. Students can observe, think and try independently according to their own ideas by staring at the dynamic elliptical trajectory repeatedly. Especially when teachers patrol among students, they can also see the way most students deal with problems when doing mathematics. 3. When the students experience observation, thinking and trying for a period of time under the guidance of the teacher, the teacher uses the method of drawing an ellipse in the textbook to demonstrate, with the purpose of further guiding the students: "When the students see that the sum of the distances from a moving point to two fixed points on the blackboard is equal to a fixed-length relationship", they can reflect the animation process just observed in their minds, so that the teacher can inspire the students to discover and understand the definition of an ellipse. 4. Let students experience the teaching situation in the problem situation. The students began to do it. The teacher walked among the students and saw the students' way of thinking in doing mathematics. Through the question and answer with the students, "Did you find the law?" Gradually, the teacher found what the teacher wanted to see.
The syllabus (including teaching time, teaching steps/processes) records the step time.
Chapter 65438 +0 ellipsis in life 1. 1 ~ 1.5 Examples of ellipsis in daily life stimulate learning motivation. 1. 1 In "Nine Planets Know All", the value of eccentricity can be changed by adjusting the position of the sun. By comparing with the numerical table nearby, it can be found that the orbit of the earth around the sun is a known ellipse, but I didn't expect it to be so close to the circle. 1.2 The conical glass is uniform, and its cross section is marked as ellipse, which can lead to other cross-section curves. 1.3 water pipes can also be oval across the board. Take a fixed length rope, fix both ends on the table with thumbtacks, and draw an ellipse by hand. 15 minutes.
Chapter II Definition of Ellipse 2. 1 ~ 2.4 Introduce the definition of ellipse formally, and explain it through computer drawing to help students understand it. 2. The definition of1ellipse comes from "drawing an ellipse with lines". 2.2 Introduce the standard formula of elliptic equation and several simple examples of finding ellipse. 2.3 and 2.4 are the definitions of conic curves in space, which were discovered in 350 BC. This is a supplementary introduction to increase students' concept of space. 30 minutes
Chapter 3 Equation of Ellipse 3. 1 ~ 3.2 Take ellipse as an example to get familiar with the solution of equation 10 minutes.
Chapter IV Point Trajectory 4. 1 ~ 4.9 It is often difficult for students to understand the meaning of the problem or how to start when solving the trajectory equation. The main reason is the lack of actual feeling and experience of how the moving point moves. Here, teachers can draw the desired trajectory for students through the computer, so that students can directly feel the formation of the ellipse through vision, see how the trajectory of the moving point is generated, and increase the experience of connecting the moving points. 50 minutes
Chapter 5 Parametric Formula of Ellipse 5. 1 ~ 5.6 introduces the parametric formula of ellipse, and makes the figure of ellipse by parametric method. Solving the extreme value problem of ellipse with parameter formula. Teachers can guess where the extreme value may appear through computer experiments and cultivate students' mathematical feeling ability when facing problems. 50 minutes
Chapter VI Ellipse of * * * Focus Through computer operation, students can understand that there can be infinitely many ellipses of * * * focus, as long as the length of long axis and short axis keeps a fixed square difference relationship. 10 minutes
Chapter VII Eccentricity of Ellipse 7. 1 ~ 7.3 When students study parabola, they know that the point with the same distance from the focus and the directrix is a parabola, but if there is a fixed ratio relationship between the two distances (between 0~ 1), they can get an ellipse. The eccentricity of ellipse is introduced, and the basic eccentricity problem is solved by using eccentricity as ellipse. The application of eccentricity can introduce the trajectories of planets and satellites. 20 minutes
Chapter 8 Properties of Ellipse 8. 1 ~ 8.3 introduces the properties of ellipse related to its tangent. In particular, the proof of 8.3 ellipse area is not a strict proof, but a compression method, which is obtained according to the proportion of the circle. In order to let students feel why the ellipse area can be found, its value is 20 minutes.
Chapter 9 Optical Properties of Ellipse 9. 1 ~ 9.5 The optical properties of ellipse come from life cases. The optical properties of ellipse are introduced. 9. 1 Understanding the optical properties of an ellipse, the light emitted from one focus will return to another focus after being reflected by the ellipse. 9.2 A question arises: "What is the continuous reflection trajectory of a bullet in an ellipse without passing through the focal point? There are two possibilities, the teacher can operate and the students can watch. 9.4 ~ 9.5 Fingers generate water waves at the focal point in the oval water tank. If we look at a single water wave, we can find that each water molecule also follows the optical properties of an ellipse. Ask the students to guess the traveling wave shape after water wave reflection. (The shape of an eye cake) Explain in detail the movement process of each water molecule. 50 minutes
Chapter 10 Appreciation of Elliptic Drawing 10. 1 ~ 10.3 Appreciation of Elliptic Drawing In addition to understanding many properties and solutions of ellipses, this chapter also introduces many other drawing methods of ellipses, which enables students to increase their knowledge of geometric drawing of ellipses, appreciate beautiful circles surrounded by straight lines and circles, and increase the number of ellipses made by projection.
1 1 interesting ellipse11. 1 ~1.3 interesting ellipse1.1elliptic connecting rod, introducing ellipse in machinery. 1.3 curvature circle of ellipse (closed circle). The definition and solution of curvature circle are not emphasized here. We tell students that there is a circle that is very close to an ellipse. An increase of one point is too big, and a decrease of one point is too small. Let students know that learning is not limited to textbook knowledge, and there are many interesting worlds waiting for you to discover and experience! 20 minutes
The analysis of students' academic performance compares the effectiveness of general teaching method and dynamic geometric drawing teaching through questionnaire survey and oral interview, and the following analysis is made: (1) Students' interest in learning has greatly increased. Nearly 90% of the students are learning through dynamic drawing for the first time. One of them said with an excited expression and tone after class: "I didn't expect that such an activity could be held in math class. The original seemingly rigid and boring graphics actually came alive on the computer!" Makes me want to learn more about this chapter. Obviously, conic curve is no longer so strange and terrible for students. (2) Through computer operation, students can have more specific and intuitive concepts. In the past, the teacher always spent a lot of time drawing various figures. Simple plane graphics are easy, of course, but when encountering more complex trajectories and three-dimensional graphics, you often need superb drawing skills, plus the drawing ability of teachers and the imagination of students. Without one of them, students' intuition will naturally be greatly reduced. Students who accept the teaching of dynamic geometric drawing can obviously understand what was previously considered abstract. (3) Learning mathematics is no longer the operation of defining and proving theorem algebra. In the process of dynamic geometric drawing, students' eyes move along the track, and images are naturally printed in their minds. Students can observe the definitions of various conic curves through graphics instead of reciting cold definitions. The teacher further guides students to think about the connection between geometry and algebra, which is a good start. I believe students will be more interested in learning other characteristics. (4) Teaching time can be more flexible. The integration of information technology into teaching not only saves time and effort, but also allows students to concentrate on thinking and reasoning, increase the interaction time between teachers and students, and really learn mathematical concepts in depth. In addition, students can also use their spare time to study by themselves and broaden their interests according to their personal needs. (5) Stimulate stronger learning motivation and desire. After understanding the definition, properties, application of optical properties and even the mysteries of the nine planets, the students enthusiastically discussed whether these concepts can be applied to daily life. Although some of them are not new or even conclusive, the discussion process has brought students a brand-new learning experience and injected new vitality into mathematics teaching and learning.
Conclusion 1. The improvement of computer mathematics software function and the progress of network technology enrich classroom teaching, so that our students not only "do" and "use" mathematics in mathematics learning activities. 2. This teaching method can provide a platform for students' diversified learning and personalized learning. We design teaching activities by improving students' learning style, and find that the same teaching content, due to different presentation methods, different teaching methods of teachers, different ways of teacher-student interaction, and different teaching content, enables students to carry out elliptical inquiry and creative learning more independently. At the same time, students are usually very excited, because they have more time to "do" math by themselves than traditional ellipse teaching, which will have a long-term impact on students' minds, rather than passively accepting the definition of ellipse. 3. Through the presentation of 3D effect of cosmo player software, the ellipse perceived by students is no longer just a set of points on the plane (rigid ellipse on the blackboard). From the perspective of three-dimensional space, an ellipse can be a straight cone or a section between a cylinder and a plane. Students can observe it from all directions and angles. The visual image of animated graphics can attract students' attention.
Other related materials 1. Teaching handout 1.doc Teaching handout 2.doc can be printed to students 2. The web pages are placed in the discoverellipse folder, and the contents of the web pages alternately used by computers and blackboards during teachers' teaching have been put on the website: http://Steiner.math.nth.edu.tw/Ne01/tyy.
Please limit this brief introduction to 3000 words. Other written or multimedia information should be attached and sent to the organizer.