Senior High School Mathematical Conic Curve I: Teaching Research on Senior High School Mathematical Conic Curve
Conic curve is an important and difficult point in high school mathematics teaching. Every year, the college entrance examination involves conic curves in various forms, including multiple-choice questions and fill-in-the-blank questions with low scores and big questions with high scores. However, students' scoring rate is generally not high. Conic curve teaching is comprehensive and systematic. This not only requires students to understand the most basic knowledge points and improve the speed and accuracy of operation, but also requires students to flexibly use the method of combining numbers and shapes to find a breakthrough in solving problems.
First, the status of high school mathematics conic teaching
1. From the teacher's point of view
The teaching objectives, important and difficult knowledge of conic curve are very clear in the senior high school mathematics syllabus. Most teachers understand the importance of conic, and they are very clear when explaining the knowledge points and problem-solving ideas of conic in class. However, students have different mathematical foundations. Some students are easy to accept the content of conic. Some students find it hard to accept. This requires teachers to pay attention to cultivating students' interest in learning in the teaching process, rather than relying solely on past teaching experience. Conic curves often use the idea of combining numbers and shapes. Some teachers tell students to use the method of combining numbers and shapes in teaching, but they don't explicitly tell students how to use this problem-solving idea. Teachers should let students know why and why. Many students can't draw inferences from one another because they study conic curves.
Considering that conic curve knowledge accounts for a large proportion in the college entrance examination, almost every year the college entrance examination questions will be involved. Therefore, teachers should consciously infiltrate in the teaching process to let students know the significance of learning conic knowledge. Conic curve is closely related to the mathematical knowledge of other modules such as vector and probability. In the teaching process, teachers should also attach importance to students' mastery of other modules' mathematical knowledge, so as to improve the efficiency of conic teaching from a macro perspective.
2. Analysis from the perspective of students
The learning of conic requires students' mathematical abilities, such as mathematical operation ability, reasoning ability and logical thinking ability. For many students, conic curves are difficult to learn. Some students are afraid of this knowledge, and the ideological burden leads to increased learning difficulties. Some students' learning methods are backward. In the process of learning, they just memorize the relevant concepts and conclusions of quadratic curve, or imitate the teaching materials and teachers' problem-solving ideas, but they don't really understand the meaning of concepts and conclusions, and they don't grasp the internal relationship between knowledge, especially their ability to comprehensively use knowledge is not enough, so they won't draw inferences. There are many kinds of conic problems. Teachers usually explain each problem in detail in class, but some students don't summarize it in time.
Second, measures to improve the teaching efficiency of high school mathematics conic section
1. Cultivate students' interest in learning conic curves.
As we all know, interest is the best teacher Students who really love learning conic curves can get twice the result with half the effort. Therefore, teachers should use effective methods to stimulate students' interest in learning conic curves. For example, in classroom teaching, teachers can create problem situations as classroom introduction. Students learn about the orbit of artificial earth satellite in the news, and teachers can introduce the knowledge of conic curve from this. When students discover the application of conic knowledge in life, their interest in learning will be greatly improved.
2. Teachers should attach importance to demonstrating the formation process of mathematical knowledge.
The multiple-choice questions and fill-in-the-blank questions in the exam do not require students to present the problem-solving process in detail, no matter what problem-solving methods are used, as long as the results are correct. But for the big questions in the test paper, the problem-solving process is very important, and a clear problem-solving process is the key to scoring, especially for the big questions of conic curves. Therefore, teachers should not only pay attention to the results, but also explain the steps of solving problems from all aspects, so that students can master the knowledge of conic curves through clear demonstrations. Move more? Many students don't know how to understand this problem. At this time, teachers should demonstrate and let students know how to use parametric solution and how to draw pictures.
3. Adhere to the students' dominant position
In teaching activities, teachers are the leaders and students are the subjects. Under no circumstances can students' dominant position be weakened. In the teaching process, teachers should understand students' cognitive laws, encourage students to explore, and let students integrate into the classroom with strong interest; Teachers should affirm and praise students more and improve their initiative and enthusiasm in learning. For some conic problems, there is more than one solution. For these questions, teachers should cultivate students' ability to explore independently, compare different solutions, and use methods with high accuracy and speed in the exam.
Third, the conclusion
Conic curve in senior high school is very difficult, so teachers should grasp the important and difficult points in teaching step by step, avoid rushing for success, and improve the difficulty on the premise of ensuring students' solid foundation. In the process of conic teaching, teachers should teach students in accordance with their aptitude, plan the teaching progress and difficulty according to students' acceptance ability, and answer students' questions patiently and seriously. Teachers should also pay attention to cultivating students' thinking of combining numbers with shapes, so as to improve the efficiency of conic teaching.
Senior High School Mathematics Conic Curve Examination Paper II: Reflections on Conic Curve Learning
According to the problems encountered in teaching, this paper tries to use the relevant knowledge of mathematical educational psychology to analyze the problems and characteristics of students' learning ellipse, and analyze the possible reasons, and then migrate to the learning process of hyperbola according to these characteristics.
Ellipse; Hyperbola; similarity
When students study ellipses and hyperbolas, teachers may pay more attention to the common problems in students' study. Although these problems are one of the factors that lead to students' learning difficulties, I think they are common among students, so it can also be considered as a kind of * * * when learning this part of knowledge. To sum up, there are mainly the following points:
1, the first definition of ellipse is too deep in memory, even a little mechanized, so that the first definition of hyperbola to be talked about later is unclear and easy to forget? Absolute value? The role of, or right? A branch of hyperbola? Or? Two? Deeply confused.
2. In deriving the standard equation of ellipse, it is difficult for students because of the large amount of calculation, although the skill of quadratic is not used. According to my statistics, nearly half of the students can't deduce the results themselves.
3. I am a little confused about the standard form required by the textbook, because the algebraic expression form appears after the second time, which should be said to be a better form. Why do you want to gild the lily and write it in fractional form?
4. When students study the geometric properties of ellipses, they feel that it is easy to find and the conclusion is beautiful, but it is difficult to remember and changeable. When they use it, they can't remember it, just remember it, don't know which attribute to use, and can't use it flexibly. Some students even think it's too magical to touch.
5. After learning hyperbola, students can find that there is a close relationship between ellipse and hyperbola, and the calculation problems of ellipse and hyperbola are similar in the process of solving, but it is generally felt that hyperbola is much more difficult than ellipse.
Although I have learned some basic knowledge of pedagogy and psychology during my undergraduate education, I have little contact with the field of educational psychology. 20 10 studied in Beijing normal university, and the hospital gave us a teacher in Xinjiang class? Mathematics education psychology? This course, the time is very short, the class hours are tense, and the learning is very shallow. But I still want to try to analyze the above problems with the help of relevant knowledge of mathematics educational psychology.
First, define ellipse and hyperbola.
? Definition? It belongs to the teaching of concepts. Mathematics education psychology? Related? Concept? Concept refers to the research object of philosophy, logic, psychology and many other disciplines. A concept usually includes four aspects: its name, definition, instance and attributes. Because the research object of mathematics is the quantitative relationship and spatial form of things, which is divorced from the specific attributes of things, mathematical concepts have corresponding characteristics. The students' cognitive structure is in the process of development, and the mathematical cognitive structure is relatively concrete and simple, and their mathematical knowledge is relatively poor. What should they do when learning new mathematics knowledge? Fixed point? There is often little or no existing knowledge.
For example, what is the definition of junior high school students' learning circle? Is the distance from the vertex on the plane equal to the trajectory of a fixed length point? At this time, only one fixed point is involved, and the fixed length is called? Radius? . The first definition of ellipse and hyperbola involves two fixed points, and? The sum of distances? With what? What is the absolute value of the distance difference? problem It is easy to think of an ellipse from the figure of a circle, but a hyperbola is more difficult. Although I learned inverse proportional function in junior high school, it is also difficult to relate it to hyperbola. In fact, this is the so-called? Experience? It is one of the influencing factors of concept learning.
Secondly, about simplifying the equation by quadratic method.
Simplify when deriving the standard equations of ellipse and hyperbola? In this process, is there a checkpoint that must be passed? Secondary leveling method? In order to achieve the purpose of removing the root sign. This method should be a necessary mathematical skill for students.
Mathematical skills are the central link from mastering mathematical knowledge to the formation and development of mathematical ability, which are divided into? Intelligent skills? And then what? Motor skills? And then what? Computing skills? Refers to the correct use of various concepts, formulas, mathematical operation rules, algebraic transformation, etc. Used correctly in this process? Mathematical symbolic language? It is also essential. In the process of mathematics learning, the formation of mathematics skills is very important. Mathematical skills are gradually formed through practical operation, learning mathematical knowledge and gaining action experience.
According to students' learning experience, in the past, there were many linear equations, and the complex quadratic function only appeared quadratic with one letter. But in the elliptic equation, the time of x and y is quadratic, which is difficult in form and difficult for students to accept psychologically. In addition, although students will use the flat method to remove roots, it is limited to the first square. The quadratic flattening method like this is not suitable, and they even suspect that they have done something wrong. In addition, because our school is a key middle school in the autonomous region and the students are excellent, it is also a factor that teachers overestimate the students' foundation and ability in teaching.
Finally, the related properties of ellipse and hyperbola.
In teaching, I found that because the first and second definitions of ellipse and hyperbola are similar, students can already feel that their geometric properties should also be similar. I also try to guide students to draw the relevant properties of hyperbola by analogy with the geometric properties of ellipse, and guide students to think spontaneously? Migration? , but for relatively simple and general, students can launch it themselves. For example: the special triangle in the ellipse, the focal radius of the ellipse, the path of the ellipse, etc. For a slightly more complicated nature, students are somewhat helpless.
Through the study of mathematics educational psychology, I found that the transfer of mathematics learning is not automatic, but influenced by many factors, the most important of which are the factors of mathematics learning materials, the generalization level of mathematics activity experience and the set of mathematics learning.
1, migration needs to analyze and abstract the old and new learning experiences and sum up the same experience components. Therefore, the mathematics learning materials should be almost the same objectively. Psychological research shows that the degree of similarity determines the effect and scope of migration.
For example, there are two fixed points and a fixed length in the definitions of ellipse and hyperbola, and the related properties of ellipse special triangle and focal radius formula derived from these conditions make it easier for students to analogy to hyperbola. It can also be found that there is only one formula for the focal radius of the ellipse, and the hyperbola should be based on the specific situation (left and right branches; The upper and lower branches are treated differently.
For another example, one of the geometric properties of an ellipse is that if a straight line passing through the focus F of the ellipse intersects with the ellipse at two points P and Q, A is a vertex on the long axis of the ellipse, and the ellipse directrix corresponding to the connecting line AP and AQ intersects with the focus F at two points M and N, then MF? NF； This property is long to describe, and students may intuitively think that the similar property of hyperbola cannot be deduced. In fact, as long as teachers give students some courage and encourage them to make bold guesses, it is easy to draw the following conclusions: If the hyperbola focus F is set to two points, P and Q, A is a vertex on the long axis of hyperbola, and the hyperbola directrix corresponding to the focus F is connected with AP and AQ at two points, M and N, respectively, then MF? NF. Then make a graphic proof. It can be said that the essence of ellipse and double thinking is very similar. 2. The transfer of mathematics learning is the influence of one learning experience on another, that is, the concretization of existing experience and the classification process of new topics or the coordination process of old and new experiences. Therefore, the lower the generalization level, the smaller the migration range and the worse the effect; On the contrary, the greater the possibility of migration, the better the effect.
For example, when exploring the geometric properties of ellipses, there is one thing: the circle with focus chord diameter PQ must be separated from the corresponding directrix; By analogy with this property, students can get that the circle with focus chord diameter PQ in hyperbola must have some relationship with the corresponding directrix. There are three positional relationships between a circle and a straight line: intersection, separation and tangency. There are two common methods to judge the positional relationship between a circle and a straight line: one is to judge by the distance from a point to a straight line; One is to judge by the root of the equation. All these knowledge and skills are possessed by students, so it is not difficult to get the related properties of hyperbola, that is, the circle with the diameter of focus chord PQ must intersect the corresponding directrix.
3. Stereotype is a kind of preparatory reaction or preparation for reaction, which occurs in continuous activities. During the activity, the experience of the previous activity forms a state of preparation for the later activity. It makes students tend to react in a specific way when studying. Because stereotype is a tendency to choose the direction of activities, the influence of stereotype can both promote and hinder migration.
For example, in the concept of ellipse, it is said that the sum of the distances to two fixed points is the trajectory of a fixed-length point, while hyperbola is the trajectory of a fixed-length point as the absolute value of the difference between the distances to two fixed points. It's easy to let go because of the mindset? Absolute value? Forgotten, so lost a hyperbola.
In view of my limited study, the analysis may not be very accurate, so I will think repeatedly in the future teaching and gradually improve it.
From the above analysis, I think there are many similarities between ellipse and hyperbola. According to students' learning characteristics, to grasp these commonalities, in addition to rich teaching experience, if teachers can also use certain psychological knowledge to discover students' psychological activities in the learning process, it may bring better teaching results.
Today, with the promotion of quality education and the implementation of a new round of national basic education curriculum reform, only pay attention to teachers? How to teach? Obviously, it is far from enough to have problems, so it is very urgent and necessary to study and discuss new textbooks and students' new learning methods. Only by giving full play to the function of mathematics education and improving the mathematics literacy of the younger generation in an all-round way can every mathematics teacher contribute to improving the quality of the whole nation and bringing up a new generation of high-quality talents.
refer to
[1] Cao,. Psychology of Mathematics Education [M]. Beijing: Beijing Normal University Press, 2007.
[2] Zhu Zhu. Middle school students' mathematics learning psychology [M]. Zhejiang Education Press, 2005.
[3] ISBN 978-7-107-18662-2, Mathematics [S]. People's Education Press, 2008.
Conical Curve of Mathematics in Senior High School 3: On the Existence of Conical Curve of College Entrance Examination
Under the guidance of the new curriculum standard, new syllabus and new exam instructions, the form and content of analytic geometry in college entrance examination have changed significantly compared with the previous syllabus. These problems have become the focus and hot spot discussed by experts and teachers, and also a test field for the reform of college entrance examination proposition. This paper probes into the problems existing in analytic geometry test questions in college entrance examination in recent years, so as to reveal how these test questions implement curriculum standards.
Keywords:: Curriculum standards, college entrance examination, thinking about the existence of mathematical analytic geometry
order
In recent years, the frequency of existential problems in college entrance examination questions is very high. Existentialism is open and divergent. The conditions and conclusions of this kind of questions are incomplete, which requires students to observe, analyze, compare and summarize in combination with the existing conditions. It requires a lot of mathematical thinking, mathematical consciousness and the ability to comprehensively apply mathematical methods, especially the second problem of analytic geometry. Is there such a view? Whether there are problems of fixed value, fixed point, straight line and circle. I hope it can provide useful thinking for teachers' teaching and college entrance examination review. [ 1]
First, is there such a constant?
Example1:(Fujian Science in 2009) It is known that AB is the left and right intersection of the curve and the axis, the straight line I passes through the point B and is perpendicular to the X axis, and S is the point on I which is different from the point B, connecting the as intersection curve C and the point T. 。
(i) If curve C is a semicircle and point T is the bisector of arc AB, try to find the coordinates of point S;
(2) As shown in the figure, point M is the intersection of a circle with a diameter of SB and a line segment TB. Is there a line that makes O, M and S three points * * *? If it exists, find the value of a, if it does not exist, please explain the reason.
Second, is there such a point?
Proposition: The second question is an exploratory and open question, and it is difficult to judge whether there is a fixed point that meets the problem. To solve this problem, we must break through two keys: one is to judge from the geometric characteristics of the figure that if there is a fixed point, it must be on the axis, and the other is to ask questions. What is the constant intersection m of a circle with PQ as its diameter? It should be translated into. Do M and K satisfy a certain relationship? Here, a certain relationship means that l is tangent to the ellipse. This topic mainly examines the ability of solving problems, reasoning and argumentation, the idea of transformation, the idea of combining numbers with shapes, and the special and general ideas. The highlight of this topic is to reflect the role of algebraic method in solving geometric problems, and at the same time to reflect the role of geometric properties of graphics in reducing algebraic operation direction and calculation amount. In reasoning and argumentation, different ways of thinking lead to different methods of solving problems, which plays a very good role in distinguishing students with different mathematical thinking levels.
3.is there such a straight line?
Proposition: The second question is an open question. Judging whether there is a straight line that meets the problem is considered from the perspective of logical thinking. Assuming that a straight line L exists, L should satisfy three conditions (1) (k can be found); (2) L and ellipse have a common point (the unequal relation between K and B can be established); ③ The distance between L and OA is equal to 4 (the equal relationship between K and B can be established), and only two straight lines are needed to determine a straight line.
So you can use L to satisfy two of the conditions, and then test the third condition to see if L exists. In this way, there are many different solutions to this problem. This topic mainly examines the ability of solving problems, reasoning and argumentation, the idea of combining functions with equations, the idea of combining numbers with shapes, and the idea of transformation. The highlight of this problem is that the background students are familiar with it, and the entrance of the test questions is wide, which can be solved with different ideas and solutions.
4. Is there such a circle?
Proposition conception: this topic belongs to exploring whether there is a problem. This paper mainly examines the determination of elliptic standard equation, the positional relationship between straight line and ellipse, the positional relationship between straight line and circle, and the method of solving equation by undetermined coefficient method. It can use the method of solving the equation to study the related parameter problems and the relationship between the roots and coefficients of the equation.
Conclusion: 1 Thinking from the teaching point of view: In teaching, we should teach the basic knowledge of straight lines, circles and conic curves and their geometric properties. In teaching, students should first intuitively understand the geometric meaning of the geometric problem to be solved through drawing, and then turn it into an algebraic problem. Through this process, students can easily understand the idea of combining numbers with shapes and the method of analytic geometry. When learning conic curves, we must first understand the geometric meaning of curve equations and parameter variables. On this basis, we should use algebraic equations to solve geometric problems. After solving geometric problems, we should return to the understanding of geometric meaning. Geometry is the starting point and destination of solving problems. It is necessary to avoid letting students fall into algebraic constant deformation without understanding its geometric meaning. We should highlight geometric features when analyzing and solving problems. We should attach importance to the algebra of geometric features and carry out algebraic identity deformation under the guidance of geometric features, so that geometric figures can help us think about problems, determine the direction of identity deformation, simplify calculations and realize the benefits brought by geometric intuition.
2. Thinking from the perspective of preparing for the third year of senior high school: ① Carefully study the exam outline and exam instructions, and clarify the requirements of the college entrance examination for the basic knowledge, basic skills, basic ideas and basic methods of analytic geometry, so that the review work can be targeted; ② Pay attention to the training of general methods of solving analytic geometry problems. From the analysis of the test questions, we can see that linear equation, circular equation, conic equation and basic properties (basic quantities) are the key knowledge points, so we must be familiar with the basic methods, and the positional relationship between straight lines and conic curves and various problems caused by them are the hot spots of subjective questions. Through the operation and explanation of typical examples, help students summarize the thinking of solving problems, think about strategies and transfer methods. In addition, we should pay attention to the intersection of analytic geometry and other mathematical contents, strengthen the cognition of knowledge integrity, and exercise students' calm psychology and strong perseverance when dealing with parameters and facing the deformation of complex mathematical formulas;
References:
[1] formulated by the Ministry of Education of People's Republic of China (PRC) and China. Mathematics Curriculum Standard of Ordinary Senior High School (Experiment) [M]. Beijing: People's Education Press, 2003
[2] Fujian Education Examinations Institute.2012 Description of Fujian Mathematics Examination for the National Unified Examination for Enrollment of Ordinary Colleges and Universities [M]. Fujian: Fujian Education Press 20 12
[3] Wang Shangzhi. Mathematics teaching research and cases [M]. Beijing: Higher Education Press, 2006.