Model essay on teaching plan of conic curve in senior high school mathematics
First, the analysis of teaching content
The definition of conic curve reflects the essential attribute of conic curve, which is highly abstract after countless practices. When solving problems properly, simplicity can control complexity in many cases. Therefore, after learning the definitions, standard equations and geometric properties of ellipse, hyperbola and parabola, we should emphasize the definition again and learn to skillfully use the definition of conic curve to solve problems. "
Second, the analysis of students' learning situation
Students in our class are very active and active in classroom teaching activities, but their computing ability is poor, their reasoning ability is weak, and their mathematical language expression ability is also slightly insufficient.
Third, the design ideas
Because this part of knowledge is abstract, if we leave perceptual knowledge, it is easy for students to get into trouble and reduce their enthusiasm for learning. In teaching, with the help of multimedia animation, students are guided to find and solve problems actively, actively participate in teaching, find and acquire new knowledge in a relaxed and pleasant environment, and improve teaching efficiency.
Fourth, teaching objectives.
1. Deeply understand and master the definition of conic curve, and can flexibly apply the definition to solve problems; Master the concepts and solutions of focus coordinates, vertex coordinates, focal length, eccentricity, directrix equation, asymptote and focal radius. Can combine the basic knowledge of plane geometry to solve conic equation.
2. Through practice, strengthen the understanding of the definition of conic curve and improve the ability of analyzing and solving problems; Through the continuous extension of questions and careful questioning, guide students to learn the general methods of solving problems.
3. With the help of multimedia-assisted teaching, stimulate the interest in learning mathematics.
Five, the teaching focus and difficulty:
Teaching focus
1. Understand the definition of conic.
2. Using the definition of conic curve to find the "maximum"
3. "Definition method" to find the trajectory equation
Teaching difficulties:
Clever use of conic definition to solve problems
Sixth, the teaching process design
Design concept
Ask questions straight to the point
As soon as I started the class, I gave it directly to-
Example 1: (1) Given that the moving point m of A (-2,0) and B (2 2,0) satisfies |MA|+|MB|=2, then the trajectory of point M is ().
(a) Ellipse (b) Hyperbolic (c) Line segment (d) does not exist.
(2) Given that the moving point M(x, y) satisfies (x 1)2(y2)2|3x4y|, the trajectory of the point m is ().
(a) ellipse (b) hyperbola (c) parabola (d) Two intersecting straight lines
Design intent
Definition is a logical method to reveal concepts. Familiarity with different definitions of different concepts is a necessary condition for learning and studying mathematics. After a period of study, students have a certain understanding of the definition of conic. Whether they can really grasp their essence is the first question I want to find out in this class.
In order to deepen students' understanding of the definition of conic, I have carefully prepared two exercises focusing on the application of the definition of conic.
Presupposition of learning situation
It is estimated that most students can answer the correct answer quickly, but some students may not really understand the definition of conic. Therefore, after the students answer, I will ask the students to say: If the answer is other options, how can the conditions be changed? This is not difficult for students who have studied this part of conic curve. But the problem (2) may be confusing for students-if a student puts forward that the problem can be solved by deformation, then I can deform the original equation according to his idea: (x 1)2(y2)2.
In this way, the correct results can be obtained soon. Otherwise, I will inspire them to learn the formula |3x4y|5 at both ends of the equation.
First of all, consider transforming it into two familiar distance formulas through appropriate deformation.
After judging the students' answers, I expanded the question to: the central coordinate of hyperbola is, the length of real axis is, and the focal length is. Deepen the understanding of the concept.
(2) Understand the definition and solve problems.
Example 2 (1) It is known that the moving circle A crosses the center of the fixed circle B: X2Y26x70, and is inscribed with the fixed circle C: XY6x9 10, so as to find the maximum value of △ABC area.
P(-2) Under the condition of (1), given point p (-2,2), find |PA|
Design intent
It is a common problem in analytic geometry to transform the problem into the mode of finding the maximum (minimum) value in geometry by using the quantitative relationship in the definition of conic curve, and it is also a kind of problem that students are easy to confuse. Example 2 is set to facilitate students' analysis.
Presupposition of learning situation
According to past experience, most students seem to be able to solve this problem smoothly, but there may not be many students who can really answer it completely. In fact, the key to solve this problem is to write the trajectory of point A accurately. With the foreshadowing of exercise 1, this problem is quite simple for students, so most students should be able to give an accurate answer to Example 2( 1), but for such a relatively unfamiliar problem as Example 2(2), students have no way to start. I remind students to associate 3/5 with eccentricity, so it is easy to associate with the second definition and find a breakthrough to solve this problem.
(C) independent exploration to deepen understanding
If time permits, these exercises will provide students with an opportunity to guess and experiment with mathematics.
Exercise: Let point Q be the minimum value of circle C: (x 1) 2225 | AB |. The moving point on 3y225, point A (1, 0) is a point on the circle, and the perpendicular lines of AQ and CQ intersect at point M. Find the trajectory equation of point M.
Extension: If point A is moved out of circle C, what will be the trajectory of point M?
The purpose of design intention exercise is to provide a platform for students to explore independently after class. Of course, if class time permits,
With the help of multimedia courseware, students can be guided to verify their conclusions.
Knowledge link
(A) the definition of conic section
The first definition of 1. conic curve
2. Unified definition of conic section
(B) Examples of the application of the definition of conic section
x2y2
The two focal points of 1. hyperbola 1 are F 1 and F2, and p is a point on the curve. If the distance from p to the left focus F 1 is 12, find P 169.
Distance to the right directrix.
|PF 1||PF2|2。 P is a point on the equilateral hyperbola x2y2a2, F 1 and F2 are two focal points, and O is the center of the hyperbola, so |PO|
Range.
3. There is a point A(4, m) on the parabola y22px, and the distance from the point A to the parabola focus F is 5. Find the parabolic equation and the coordinates of point A. ..
x2y2
4.( 1) It is known that point f is the right focus of ellipse 1, m is the fixed point on this ellipse, and a (2 2,2) is the fixed point. Found 259.
The minimum value of |MA|+|MF|.
X2y2 1 1(2) It is known that a (,3) is a certain point, f is the right focus of hyperbola 1, and m moves on the right branch of hyperbola. When 9272
When 1|AM||MF| is minimum, find the coordinates of point m.2.
x2
(3) Given the point P (-2,3) and the parabola Y with the focus f, find the point M on the parabola to minimize |PM|+|FM|. eight
x2y2
5. It is known that A (4 4,0), B (2 2,2) are the points on the ellipse 1, and M is the moving point on the ellipse. Found the 259th of |MA|+|MB|.
Minimum and maximum values.
Seven, teaching reflection
1. With the help of "www.liuxue86.com", this course will make it possible for all students to participate in activities, and make abstract mathematical theories that were originally difficult to understand vivid, vivid and easy to understand. At the same time, the use of "multimedia courseware" to assist teaching will save the time of performance, thus leaving more time for students to carry out self-enlightenment, self-training and self-examination, giving full play to students' main role and fully embodying "multimedia courseware"
2. Using two examples and their extensions, through a changeable topic, in-depth exploration at different levels, detection and research on the guessing results, cultivate students' thinking ability, enable students to learn to solve a problem and gradually master the solution of a class of problems. Combine two kinds of "maximum problem" that students are easy to confuse into one problem, which is convenient for students to compare and analyze. Although on the surface, my teaching capacity is not large, in fact, students' thinking exercise is not small.
In a word, how to better choose examples and exercises that meet students' specific conditions and teaching objectives, and how to flexibly grasp the pace of classroom teaching are still important research topics in my future work. To really implement quality education and cultivate students' innovative consciousness, I must first update my concept-to use multimedia technology moderately in teaching, so that students have the opportunity to participate in teaching practice and stimulate their curiosity while learning new knowledge. In the process of seeking to solve problems, I gain self-confidence and successful experience, and unconsciously improve my thinking quality and mathematical thinking ability.
High school mathematics geometric series excellent teaching plan
Teaching objectives
1. Understand the concept of geometric series, master the general formula of geometric series, and use the formula to solve simple problems.
(1) Understand the definition of geometric series correctly, understand the concept of common ratio, make clear that a series is the limit condition of geometric series, judge that a series is geometric series according to the definition, and understand the concept of equal ratio term;
(2) Correctly understand the usage of the representation of geometric series, and flexibly use the general formula to find the first term, common ratio, number of terms and specified terms of geometric series;
(3) Understanding the essence of geometric series by general formula can solve some practical problems.
2. Through the study of geometric series, gradually cultivate students' thinking qualities such as observation, analogy, induction and conjecture.
3. By summarizing the concept of geometric series, we can further cultivate students' rigorous thinking habits and scientific attitude of seeking truth from facts.
Textbook analysis
(1) knowledge structure
Geometric series is another simple and common series, and its research content is comparable to that of arithmetic progression. Firstly, the definition of geometric series is summarized, and the general term formula is derived. Then, the image is studied, and the concept of equal ratio median term is given. Finally, the application of the general formula is given.
(2) Analysis of key points and difficulties
The focus of teaching is the definition of geometric series and the understanding and application of general formula, while the difficulty of teaching lies in the derivation and application of general formula of geometric series.
(1) Like arithmetic progression, geometric progression is a special series. They have many similar properties, but there are also obvious differences. According to the definition and general formula, the characteristics of geometric series can be obtained, which is the focus of teaching.
(2) Although I have been exposed to incomplete induction in arithmetic progression's study, it is still unfamiliar to students; In the process of deduction, students need to have certain ability of observation, analysis and guess; Whether the first term is established needs to be supplemented, so it is difficult to deduce the general term formula.
③ The comprehensive study of arithmetic progression and geometric progression can not be separated from the general formula, so the flexible use of the general formula is both important and difficult.
Teaching suggestion
(1) It is suggested that this course be divided into two classes, one is about the concept of geometric series, and the other is about the application of general formula of geometric series.
(2) With the introduction of the concept of geometric series, we can cite several concrete examples, and students can sum up the same characteristics of these series, so as to get the definition of geometric series. We can also give several arithmetic progression and geometric progression together. Students can classify these series, one of which is divided according to arithmetic and proportional, so that Tsuihiji can sum up the definition of geometric progression.
(3) According to the definition, let students analyze the characteristics that the common ratio of geometric series is not 0 and each term is not 0, so as to deepen their understanding of the concept.
(4) Compared with arithmetic progression's representation, students can summarize geometric progression's representation, inspire students to understand the general formula from the perspective of function, and draw the image of series from the structural characteristics of the general formula.
(5) Thanks to arithmetic progression's research experience, geometric progression's research can be solved by the students themselves. Teachers only need to grasp the rhythm of the class and appear as the organizer of a class.
(6) Students can ask questions, solve problems and give lectures to each other, and give full play to students' main role.
Example of instructional design
Title: The concept of geometric series.
Teaching objectives
1. Through teaching, students can understand the concept of geometric series and deduce and master general formulas.
2. Make students further understand the ideas of analogy and induction, and cultivate the ability of observation and generalization.
3. Cultivate students' diligent thinking, seeking truth from facts and rigorous scientific attitude.
Teaching emphases and difficulties
The emphasis and difficulty are the induction of the definition of geometric series and the derivation of general formula.
training/teaching aid
Projector, multimedia software, computer.
teaching method
Discussion and conversation methods.
teaching process
First, ask questions.
Give the following series, classify them and say the classification criteria. (slide)
①-2, 1,4,7, 10, 13, 16, 19,…
②8, 16,32,64, 128,256,…
③ 1, 1, 1, 1, 1, 1, 1,…
④243,8 1,27,9,3, 1,,,…
⑤3 1,29,27,25,23,2 1, 19,…
⑥ 1,- 1, 1,- 1, 1,- 1, 1,- 1,…
⑦ 1,- 10, 100,- 1000, 10000,- 100000,…
⑧0,0,0,0,0,0,0,…
Students express their opinions (according to the relationship between items, they may be divided into increasing series, decreasing series, constant series and swinging series, and may also be divided into arithmetic progression series and equal proportion series), and unify a classification method, in which 2346 ⑥ is a series with the same nature (it's just as well that students can't see ③, so we will check whether ③ is a geometric series after we get the definition).
Second, explain the new lesson.
Ask the students to say the same characteristics of the sequence 2346. The teacher pointed out that there are many similar examples in real life, such as the problem of amoeba division. Suppose that every unit time passes, each amoeba splits into two amoebas, and then suppose that there is an amoeba at first, and after a unit time, it splits into two amoebas, and after two unit times, there are four amoebas ... Go on, record the number of amoebas in a unit time.
This series also has the same characteristics as the last series, that is, another series we want to study-geometric series (here is the first step to play the multimedia software of the Transformers)
Geometric series (blackboard writing)
1. Definition of Geometric Series (blackboard writing)
According to the difference and connection between geometric series and arithmetic progression's name, this paper tries to define geometric series. Students' general answers may not be perfect. In most cases, students can sum up the foundation of arithmetic progression. The teacher writes the definition of geometric series and marks the key words.
Ask the students to point out the common ratio of 2346⑥ in geometric progression, and think that there are countless columns that are both arithmetic progression and geometric progression. Students can find that ③ is such a series through observation. The teacher will ask if there are any other examples. Let the students give two more examples. Then, ask the students to summarize the general form of this series. Students may say that all shapes of series satisfy both arithmetic and geometric series. Ask the students to discuss and draw a conclusion. At that time, the series was both arithmetic progression and geometric progression, and at that time it was only arithmetic progression.
2. Understand the definition (blackboard writing)
The first term of (1) geometric series is not 0;
(2) Every term of the geometric series is not 0, that is,
Question: If all the items in a series are not zero, what are the conditions for the series to be a geometric series?
(3) The common ratio is not 0.
The definition of geometric series is expressed by mathematical formula.
This is a geometric series.
The writing of this formula may be controversial, such as the writing.
Let the students study it, ok; Then ask, can you rewrite it as
Is it a geometric series? Why not? The formula gives the first term and the second term of the sequence.
Item number, but can you determine a geometric series? How many conditions do you need to determine a geometric series? Given the first term and the common ratio, how to find the value of either term? So we should study the general formula.
3. General formula of geometric series (blackboard writing)
Question: Use and to represent items.
① Incomplete induction
② Iterative multiplication
The formula,,, is multiplication, so
General term formula of (1) geometric series
After getting the general formula, let the students think about how to understand the general formula.
(blackboard writing) (2) understanding of the formula
For students, it ultimately boils down to:
① Functional view;
(2) the equation thought (because we already know it in arithmetic progression, review and consolidate it here).
The idea of equation is emphasized here to solve the problem. There are four quantities in the equation, which is the simplest application of the formula. Please give the students an example (you should be able to make up four kinds of questions). What is the format of solving the problem? (Not only to solve problems, but also to pay attention to the training of standardized expression)
If you add a condition, you will know one more quantity, which is a higher level application of the formula. We will learn it in the next class. Students can try to make up some questions.
Three. abstract
1. This lesson learned the concept of geometric series and got the general formula;
2. Pay attention to the analogy with arithmetic progression in research contents and methods;
3. Understand the general formula with the idea of equation and apply it.
Investigation activities
Fold a large piece of tissue paper in half. How thick is it after 30% discount (if possible)? Suppose the thickness of this paper is 0.01mm.
Reference answer:
After 30 times, the thickness is, which exceeds the height of Mount Everest, the highest peak in the world. If the paper is thinner, for example, the paper thickness is 0.00 1 mm, it will exceed the height of Mount Everest if it is folded in half for 34 times. Remember the king's promise? The rice in 3 1 grid is already 107374 1824, and the rice in the following grid is even more. The rice in the last grid should be 6 grains. Use a calculator to calculate (logarithmic calculation is also acceptable).
Teaching plan design of high school mathematics sequence
I. teaching material analysis
Status and function
Sequence is one of the important contents of high school mathematics, which not only has a wide range of practical applications, but also has the function of connecting the past with the future. On the one hand, sequence, as a special function, is inseparable from function thought; On the other hand, learning sequence is also a preparation for further learning the limit of sequence. On the other hand, on the basis of students' learning the concept of sequence, arithmetic progression gave two methods of sequence-general formula and recursive formula, which further deepened and broadened his understanding of sequence. At the same time, arithmetic progression also provided a foundation for studying geometric series in the future.
(2) Analysis of learning situation
(1) The students have mastered _ _ _ _ _ _ _ _ _ _ _.
(2) Students have rich knowledge and experience, strong abstract thinking ability and deductive reasoning ability.
(3) Students have lively thinking and high enthusiasm, and initially formed the ability to explore mathematical problems in cooperation.
(4) Students' different reference levels are uneven, and individual differences are obvious.
Second, the target analysis
The new curriculum standard points out that "three-dimensional goal" is a closely related organic whole, which should be a process of acquiring knowledge and skills, and at the same time become a kind of learning and correct values. This requires us to take the cultivation of knowledge and skills as the main line, infiltrate emotional attitudes and values, and fully reflect them in the teaching process. The new curriculum standard points out that the main body of teaching is students, so the formulation and design of objectives must start from the students' point of view, according to the position and role of _ _ _ in the teaching materials, and combined with the analysis of learning situation, the teaching of this class should achieve the following teaching objectives:
(A) Teaching objectives
(1) knowledge and skills
Make students understand the concept of monotonicity of function and master the method of judging monotonicity of function; .
(2) Process and method
Through observation, induction, abstraction and generalization, students are guided to independently construct concepts such as monotone increasing function and monotone decreasing function. Can use the concept of monotonicity of function to solve simple problems; Make students understand the mathematical thinking method of combining numbers and shapes, and cultivate students' ability to find, analyze and solve problems.
(3) Emotional attitudes and values
In the learning process of monotonicity of function, let students experience the scientific value and application value of mathematics, and cultivate students' good habits of observation and exploration and rigorous scientific attitude.
(2) Key points and difficulties
The teaching focus of this lesson is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
Thirdly, the analysis of teaching methods and learning methods
teaching method
According to the content characteristics of this class and the age characteristics of senior two students, and according to the classroom teaching strategy of "354" in Linyi senior high school mathematics, the inquiry-experience teaching method is adopted to complete the teaching. In order to achieve the teaching objectives of this class, I adopted the following teaching methods:
1. Introduce topics through real life problems that students are familiar with, create situations for concept learning, narrow the distance between mathematics and reality, stimulate students' thirst for knowledge and arouse their enthusiasm for participation.
2. In the process of forming the concept, closely follow the key sentences in the concept and form the concept correctly through the participation of students.
3. While encouraging students to participate, the leading role of teachers should not be ignored, and students should be taught clear thinking, rigorous reasoning and successful written expression.
(2) study law
I attach importance to studying law;
1, let students intuitively enlighten their thinking with graphics, and complete the qualitative leap from perceptual knowledge to rational thinking through the construction of positive and negative examples.
2. Let students question, try, induce, summarize and apply from problems, and cultivate students' ability to discover, study, analyze and solve problems.
Fourthly, the analysis of teaching process.
(A) Teaching process design
Teaching is a harmonious whole composed of teachers' "guidance" and students' "learning" and "enlightenment" in the teaching process. Teachers' "guidance" means that teachers inspire, induce, motivate and evaluate students' learning, and transfer learning tasks to students. Students just accept tasks, explore problems and complete tasks. If "teaching and learning" are perfectly combined in the teaching process, that is, taking "problems" as the core, we can organize and promote teaching through deduction, explanation and exploration of the occurrence, development and application of knowledge.
(1) Create a situation and ask questions.
The new curriculum standard points out: "Let students learn mathematics in concrete and vivid situations". In this class, we ask questions from familiar life situations. The design of questions has changed the traditional design method with clear purpose, given students the greatest thinking space, and fully reflected students' dominant position.
(2) Guiding inquiry and constructing concepts.
The formation of mathematical concepts comes from the need to solve practical problems and the development of mathematics itself. However, the high abstraction of concepts brings difficulties to understanding, teaching and learning, which requires students to be involved in their own practical learning activities and experience the process of "mathematization" and "re-creation" on the basis of their own experience and existing knowledge.
(3) Self-attempt and preliminary application.
Effective mathematics learning process can not be simply imitated and memorized, especially the understanding and learning process of mathematical thinking. Let students experience and practice in the process of solving problems, teachers and students learn interactively, students cooperate and communicate, and explore.
(4) Consolidation and deepening of in-class training.
Through the participation of students, students can deeply understand the main contents and thinking methods of this lesson, so as to deepen their knowledge again.
(5) Summary, review and reflection.
Summary is not only a simple knowledge review, but also a summary of knowledge, methods and experiences by giving full play to students' dominant position. I designed three questions: (1) What did you learn through this lesson? (2) What is your biggest experience from this lesson? (3) What skills have you mastered through this lesson?
(2) Work design
Homework is divided into compulsory questions and multiple-choice questions. The required questions reflect the knowledge level of students in this course. Multiple-choice questions are an extension of the content of this course, focusing on the extensibility and coherence of knowledge and emphasizing the application of what you have learned. Through homework setting, students at different levels can get the joy of success and see their potential, thus stimulating students' full interest in learning and promoting the formation of a learning atmosphere of independent development and cooperative inquiry.
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