1. Master the definition of ellipse, two forms of elliptic standard equation and their derivation process;

2. Be able to determine the standard equation of ellipse according to conditions and master the standard equation of ellipse by undetermined coefficient method;

3. Introduce the concept of ellipse into teaching to cultivate students' observation ability and exploration ability;

4. Through the derivation of elliptic standard equation, students can further master the general method of solving curve equation, penetrate the thinking method of combining numbers and shapes and equivalent transformation, and improve their ability to solve geometric problems by coordinate method;

5. Let China Learning Alliance explore the definition and standard equation of ellipse, stimulate students' enthusiasm for learning mathematics, and cultivate students' interest in learning and innovative consciousness.

Teaching suggestion

Textbook analysis

1. knowledge structure

2. Analysis of key and difficult points

The definition of ellipse and two forms of elliptic standard equation are introduced emphatically. The difficulty is the establishment and derivation of elliptic standard equation. The key is to master the method of establishing coordinate system and simplifying radicals.

The textbook Ellipse and its Standard Equation is generally composed of two parts: one is the definition of ellipse; The second is the standard equation of ellipse. Ellipse is the first of the three conic curves to be studied in this chapter, so the textbook focuses on ellipse and consolidates its application in the teaching of hyperbola and parabola. First, it is said that the ellipse is also naturally related to the equation of the circle in Chapter 7. Learning ellipse well is very important for students to learn conic well.

(1) To understand the definition of an ellipse, we must grasp the conditions that the points on the ellipse should meet, that is, the geometric properties of the points on the ellipse, which can be understood by comparing with the definition of a circle.

In addition, we should pay attention to the definition of "constant", that is, constant should be greater than. This rule is to avoid two special situations, namely: "When the constant is equal to, the trajectory is a line segment; When the constant is less than, there is no trajectory. " This will help to concentrate on further studying the standard equation and geometric properties of ellipses. However, when explaining the definition of ellipse, we should pay attention not to ignore these two special cases to ensure the accuracy of ellipse definition.

(2) According to the definition of ellipse, we should pay attention to the following points:

① The equation of the curve depends on the coordinate system, and the establishment of a suitable coordinate system is the first thing to pay attention to when solving the curve equation. Students should observe the figure or reason of ellipse according to the definition of ellipse and find that ellipse has two perpendicular symmetry axes. Taking these two symmetrical axes as the two axes of the coordinate system can not only make the derivation process of the equation simple, but also make the final form of the equation neat and concise.

② Let the focal length of the ellipse be, and the distance from any point of the ellipse to the two focal points be. All these measures are to simplify the derivation process, make the equation form neat and concise, and make it easy for students to understand it carefully.

③ The simplification of irrational equations in the process of equation derivation is not only a common problem for us to solve trajectory equations in the future, but also a difficulty for students. The simplification method of this kind of equation should pay attention to: ① When the equation has only one root, it should be left on one side of the equation alone, and the other terms should be moved to the other side; (2) When there are two radicals in the equation, they should be placed on both sides of the equation, and there is only one term on one side.

④ The derivation of the elliptic standard equation in the textbook actually only gives that "the coordinates of the points on the ellipse are all suitable for the equation" but does not prove that "the solution of the equation is that all the points with coordinates are on the ellipse". This is actually the deformation problem of the same solution equation, which is more difficult and does not require students.

(3) Similarities and differences of ellipses in the two standard equations.

The center of the circle is at the origin, the focus is on the axis, and the standard equations of the ellipse on the axis are:, where. Their similarities are: the same shape, the same size, all of them. The difference is that the positions of two ellipses relative to the coordinate system are different, and their focal coordinates are also different.

The focus of the ellipse is on the axis, and the denominator of the term in the standard equation is large;

The focus of the ellipse is on the axis, and the denominator of the term in the standard equation is large.

In addition, as long as,, have the same sign, it is an elliptic equation, which can be transformed into.

(4) In the textbook, another commonly used method for solving trajectory equation-intermediate variable method is introduced through example 3. Example 3 has three functions: the first is to teach students how to use intermediate variables to find the trajectory of a point; The second is to explain to students that if the equation form of the trajectory of a point is the same as the standard equation of an ellipse, then the trajectory is an ellipse; Third, let students know that an ellipse can be obtained by stretching a circle in a certain direction.

Teaching suggestion

(1) Make students understand the application of conic curve in production and technology, and stimulate students' interest in learning.

In order to stimulate students' interest in learning conic, and realize the role of conic knowledge in real life, we can introduce practical questions, from which we can ask questions about conic, let students know what they want to study, such as the examples in the book, and also inspire students to find examples related to conic around them.

For example, the earth we live in always revolves around the sun ellipse, and so do other planets in the solar system. The sun is located at a focus of the ellipse. If the speed of these planets increases to a certain extent, they will run along parabola or hyperbola. Humans should follow this principle when launching artificial earth satellites or artificial planets. Relative to an object, the motion of another object attracted by it is impossible to have any other orbit according to the law of universal gravitation. Therefore, in this sense, the conic curve constitutes the basic form of our universe. In addition, the contour line of the factory ventilation tower and the axial section curve of the searchlight reflector are all related to the conic curve, which is of great value in real life.

(2) After class, arrange for students to cut conical things, so that students can understand the origin of the name of conic curve.

In order to let students know the origin of the name of conic, but in order to save class time, students should be arranged to cut their own carrots and cement after class to deepen their understanding of conic.

(3) When introducing the definition of ellipse, we should pay attention to the use of intuitive and vivid models or teaching AIDS, so that students can start from perceptual knowledge and gradually rise to rational knowledge to form correct concepts.

Teachers can talk about the shadow of radish slices and disks on the ground under the sun from the orbits of the sun, the earth and artificial earth satellites, so that students can have an intuitive understanding of the ellipse first.

The teacher can prepare a thin thread and two nails in advance. Before giving a strict definition of ellipse in mathematics, the teacher first takes two fixed points on the blackboard (the distance between the two fixed points is less than the length of thin lines), and then asks two students to draw an ellipse on the blackboard according to the teacher's requirements. After drawing, the teacher takes two fixed points on the blackboard (the distance between the two fixed points is greater than the length of the thin line), and then asks the two students to draw according to the same requirements. By observing the process of drawing twice, students sum up their experiences and lessons, and the teacher guides them according to the situation, so that students can draw a strict definition of ellipse. In this way, students will have a deep understanding of this definition.

(4) The proposed problem is decomposed into several sub-problems, and the essence of ellipse definition is embodied by multimedia courseware.

Several questions can be set in teaching, so that students can use their hands and brains, think independently and explore independently, and find solutions to the problems by using multimedia through observation, experiment and analysis. In the teaching process of the definition of ellipse (), it can be put forward that "the trajectory of a point whose sum of the distances from a point to two fixed points is a constant value must be an ellipse", so that students can demonstrate "changing the focal length or constant value" through courseware and observe the shape of the trajectory, so as to dig out the connotation of the definition and impress students with the definition of ellipse.

(5) Pay attention to the relationship between the definition of ellipse and the standard equation of ellipse.

When explaining the definition of ellipse, we should inspire students to pay attention to the graphic characteristics of ellipse, so that students can easily find the symmetry of ellipse, so that when establishing coordinate system, students can easily choose the appropriate coordinate system, even if the focus is on the coordinate axis and the center of symmetry is the origin (don't study the geometric properties too much at this time). Although students may not be able to explain why they choose the coordinate system in this way, it is easier for them to accept and explain the general principle of choosing the appropriate coordinate system on the basis of a certain perceptual knowledge.

(6) Teachers should pay attention to solving difficulties when deriving the standard equation of ellipse, and supplement the method of radical simplification in time.

When deriving the standard equation of ellipse, because the sum of the listed equations is equal to a non-zero constant, it needs to be squared twice when simplifying, and there are more than three letters in the equation, and the number of times and terms are high. We should pay attention to solving difficulties in teaching, so as not to affect students' overall understanding of the derivation process of elliptic standard equations. Through concrete examples, let students use formulas to simplify the equation step by step, that is: (65433 (2) When there are two expressions in the equation, they should be placed on both sides of the equation, and only one item can be found on one side (to avoid secondary operation).

(7) After explaining the standard equation of an ellipse with the focus on the X axis, the teacher should inspire students to learn the standard equation with the focus on the Y axis, and then encourage students to explore the similarities and differences between the two standard equations of an ellipse and deepen their understanding of the ellipse.

(8) Consolidate old knowledge on the basis of learning new knowledge.

Ellipse is also a kind of curve, so we still need to use the knowledge of curve and equation mentioned in chapter 7, so we should pay attention to further consolidate the concepts of curve and equation when deriving the standard equation of ellipse. After the standard equation of ellipse is derived from the textbook, it is not proved that the equation is indeed an elliptic equation, so it should be explained to students that it is not contradictory to the concepts of curve and equation mentioned above, but because the simplification process of elliptic equation is equivalent deformation, and the proof process is complicated. Therefore, there is no requirement in the textbook and there is no proof process, but students should pay attention to it and do not need to prove it in the future. What they should pay attention to is that only the simplification of the equation is equivalent and deformed, but in fact, students encounter some specific problems and need specific analysis of specific problems.

(9) It is necessary to emphasize both the leading role of teachers and the main role of students. In class, let all students participate in the discussion as much as possible, students with poor foundation make guesses, and students with good foundation help prove it, so as to cultivate students' team spirit of unity and cooperation.