First, the presentation of teaching clips-to cultivate children's ability of geomantic omen
Paragraph 1: Review the analytic formula of quadratic function.
Teacher: What are the analytical formulas of quadratic function?
Health: general formula y = ax2+bx+c (a ≠ 0); Vertex y = a (x-k) 2+h; Intersection y=a(x-x 1)(x-x2)
Teacher (showing basic exercise 1): Knowing the image passing points of quadratic function (1, 0), (2,-1) and (3, 0), find the analytical formula of this quadratic function.
Students 1 and 2 went to the podium in turn to explain the general solution and the delivery solution.
S3: I made it with the intersection (and showed the solution on the physical projector). Because the parabola passes through points (1, 0) and (3,0), its symmetry axis is straight line x=2, and because it passes through point (2,-1), its vertex is (2,-1), so we might as well set its equation to y = a (.
Teacher: Why is the symmetry axis a straight line x=2?
Health 3: The root is two points with equal ordinate.
(Comments: At the beginning of the class, the teacher's few words activated the classroom, activated the students' thinking, and the students showed it elegantly on stage, laying the foundation for creating a good ecological classroom environment; Under the student-oriented teaching concept, various analytical formulas of quadratic function are reviewed and trained. In the comprehensive presentation and comparison of various methods, students' re-understanding of key conditions is highlighted, and the theme of this lesson "Symmetry of quadratic function" is illustrated intuitively and clearly, which strengthens the optimization consciousness of problem-solving strategies. )
Section 2: Explore the size of the function value of quadratic function.
Teacher (Show Basic Exercise 2): The given points A(- 1, y 1) and B(5, y2) are two points on the image of the function y=x2-4x+3, and the relationship between y 1 and y2 is-
Health 4: I first formulated it as y=(x-2)2- 1, and learned that the symmetry axis is a straight line x=2, and then combined with the image, I learned that y 1=y2.
Health 5: No formula is needed. I know from the conclusion of 1 that the axis of symmetry is a straight line x=2.
Students 6: y 1 and y2 are calculated by special value method.
Teacher: It is known that points A(-2, y 1) and B(5, y2) are two points on the image of function y=x2-4x+3, and the relationship between y 1 and y2 is not compared by calculation.
Health 7: Because the symmetry axis is a straight line x=2, you can know y1> from the image; y2 .
Teacher: Can you describe it in mathematical language?
Health 8: When a>0, quadratic function y=ax2+bx+c, the closer a point on the image is to the symmetry axis, the smaller its ordinate is; When a<0 and quadratic function y=ax2+bx+c, the closer a point on the image is to the axis of symmetry, the larger its ordinate is.
Teacher: Is there any other way?
Health 9: The symmetrical point of point A (-2, y 1) about the axis of symmetry is A(6, y 1), because both point A (6, y 1) and point B (5, y2) are on the right side of the axis of symmetry, and point A (6, y/kloc) y2 .
Teacher: That is to say, we can examine the distance between two points and the symmetry axis, or we can transform it to the same side of the symmetry axis.
The teacher continued to change the conditions on the basis of the variant question 1, and the following variants appeared:
If point A (X 1, 0) and point B (X2, 0) are set in Variant 2, then the value of x=x 1+x2 at that time is?
If point A (X 1, 5) and point B (X2, 5) are set in Variant 3, then the value of x=x 1+x2 at that time is?
Variable 4 When x is equal to x 1 and x2(x 1≠x2) respectively, the value of y is equal, so when x=x 1+x2, the value of y is?
Variant 5 is known that for the quadratic function y=ax2+bx+c(a≠0), the value of y is equal when x is equal to x 1 and x2(x 1≠x2) respectively, so the value of y is equal when x=x 1+x2.
(Comment: Further change the original problem, guide students from specific problems to a broader problem space, and change a single problem-solving into a method system that consolidates knowledge and forms problem-solving strategies. Through constant changes, students can constantly clarify and strengthen the core idea of this lesson: using the symmetry of quadratic function to skillfully solve the problem of the value of quadratic function. In the teacher's gradual progress, the symmetrical beauty of quadratic function is gradually highlighted. )
The third part: Explore the range of quadratic function.
Teacher (Show Basic Exercise 3): Draw a sketch of the function y=x2-4x+3 and answer the following questions:
(1) When 3≤x≤5, the value range of y is;
(2) When 2≤x≤5, the value range of y is;
(3) When 0≤x≤5, the value range of y is.
Health 10: the answers to the three questions are 0≤y≤8,-1≤y≤8,-1≤y≤8 respectively.
Health 1 1: I don't understand why the range of X in sub-items (2) and (3) is different, but the range of Y is the same. I think the answer to question (3) should be 3≤y≤8.
Health 12: It depends not only on the value of the endpoint, but also on which part of the image the graph 1 x corresponds to when it changes in a certain range, and then on the range of the ordinate of this part of the image.
Teacher: That's good! Observe the image and speak from the picture!
(Then draw the images corresponding to the three small questions. )
Teacher: if t≤x≤5,-1≤y≤8, then the range of t is teacher: if T >；; 2, and then-
All beings: y can't get-1.
Teacher: If t<- 1, then-
Sentient beings: y can also get a value greater than 8.
Teacher: If so, then-(ask questions and draw the corresponding images, the starting point of the parabolic segment slides between A and B, and the ending point is fixed at C)
Sentient beings: y can get all values ≥- 1 and ≤8, but can't get other values.
(Comments: Students ask questions independently and solve problems interactively, and teachers provide timely guidance to solve problems, giving students the maximum opportunity of independent inquiry and interactive communication, so that students can expose and solve problems. In this series of processes, students are always the protagonists. In the whole process of inquiry, students are observing images, using images and speaking from images. The starting point of thinking starts with pictures, the breakthrough of difficulties depends on pictures, and the right or wrong of conclusions depends on pictures to test. The idea of the combination of numbers and shapes was cultivated in the brains of junior three students intentionally or unintentionally. )
Second, the third teaching suggestions-forward-looking to promote convergence
1. Explore more and instill less.
The famous Swiss educator Pestalozzi said: "The main task of teaching is not to accumulate knowledge, but to develop and train thinking, which will help students expand their thinking and cultivate innovative spirit." Therefore, taking students as the main body and teachers as the leading factor, organically integrating inquiry activities into the mathematics classroom of grade three is the most effective measure to do a good job in the connection of mathematics teaching in junior and senior high schools, and it is the real way to teach people to "fish".
In the new teaching, the generation of concepts is the core and sometimes even the difficulty. We should guide students to fully explore, conduct experimental exchanges and reflections based on personal experience, and let students construct in personal experience, which can not only effectively break through the difficulties in concept teaching, but also better help students deepen their understanding of concepts and cultivate their awareness and ability to use concepts. If the teaching of theorems, formulas and rules is only mechanically memorized by students and directly applied to solving problems, it will directly lead to the students (including the top students in the senior high school entrance examination) who have extremely limited understanding ability, poor understanding, learning difficulties and plummeting grades. Therefore, teachers should moderately "reprocess" the teaching materials, give students time and space to "rediscover" and "recreate" the soil, let them explore "mathematical laws" independently, naturally generate and learn from their own psychological needs and emotional experiences, and find firm attachment points and growth points for formula theorems.
For example, in the teaching of example exercises, if we blindly "demonstrate → imitate → demonstrate → imitate", students' intuition, understanding, self-confidence and interest will be worn away, resulting in dependence, inertia, fatigue and boredom. Moreover, after high school, the amount of knowledge will increase exponentially, and it is difficult for students to cope with the good memory. Therefore, in the third grade, it is necessary to stimulate students' interest in solving problems and improve their inherent ability to solve problems as the main theme, decompose difficulties through "problem fission", guide students to explore step by step, expand and extend through analogy, guide students to explore deeply, achieve mastery through a comprehensive study, and explore clear and correct conclusions through observing images, and so on.
In the second segment of this lesson, Mr. Wu vividly embodies that "the problem is the heart of mathematics". Under the guidance of a series of close relatives' questions, students enjoy it, so they actively participate in the inquiry, enjoy the process of experiencing inquiry, feel the pleasure of success, construct a perfect questioning system and coping methods, cultivate the flexibility, flexibility and profundity of thinking, and are quite eager to try.
2. More cooperative demonstrations, less teacher performances.
If the class type of each class is fixed and the mode is routine, then the classroom will inevitably fall into boredom. At this time, it is necessary to inject "living water" into the classroom to make it smart. This "living water" is the result of students' original ecological thinking.
In this class, the students stepped onto the platform one by one, holding the pointer, which showed that although they were not as incisive as the teacher and as dazzling as the host, their language expression was clear and fluent, showing their confidence in their gestures. When students come to the stage to explain, they can sort out problems, correct mistakes and effectively avoid mistakes in after-school homework. All kinds of good ideas and ideas are displayed and exchanged in the first time, so as to realize wisdom sharing, gain success and self-confidence, and inspire "catching up with school and helping others"! This kind of energy is beyond the reach of any high-level teacher. Because of the difference in experience and cognitive level, a brain has to take care of dozens of people and really "think what students think and make mistakes", which is not small. In senior high school, with the rapid increase of the breadth and depth of mathematics learning content, the problems that can be solved in one class are limited, and more students need to solve them in extracurricular cooperation and communication. Therefore, it is of far-reaching significance to develop good cooperative communication habits in junior high school.
3. There are many ways of thinking, but few skills are shielded.
Mathematics teaching is not only a systematic combing and scanning of mathematical knowledge, but also a classification and summary through teaching, so as to master universality and generality. Once a student produces mathematical ideas and methods in class, his ability to solve problems will advance by leaps and bounds. In the process of producing new concepts and new knowledge, and in the process of solving problems, students should feel the rationality and necessity of relevant mathematical ideas, and consciously apply important mathematical thinking methods such as equivalent transformation, classified discussion and combination of numbers and shapes; Let students deeply experience the role of mathematical thinking method in the key points of teaching focus and difficult breakthrough; When students succeed, if students are allowed to strike while the iron is hot and consolidate their training, and at the end of each class, if teachers can properly lead students to summarize the knowledge and methods of this class, it will be of great benefit to enhance students' understanding and formation of mathematical thinking methods.
The doubt and difficulty in the third paragraph of this lesson lies in twists and turns, relying on the combination of numbers and shapes to make the final decision, forcing and inducing students to supplement the classified discussion on the basis of the combination of numbers and shapes. At this moment, not only has the problem been solved, but what is more precious is that the "window" of sunshine is slowly opened in the students' thinking bank-the combination of numbers and shapes, classified discussion. These are common thinking weapons in high school mathematics.
4. More point-to-point changes and less direct narration.
Educational psychology tells us that only continuous learning experience can constitute meaningful learning experience, and fragmented, decentralized and monotonous learning experience often cannot constitute meaningful learning. Therefore, necessary repetition becomes the premise of ensuring continuity, but repetition itself can easily lead to the narrowness of learning experience, which is inconsistent with the original meaning of "learning" (including the meaning of "improving"). Therefore, it is necessary to guide "transcendence" (such as providing variants).
Teachers should first select problems, let students solve problems by their own intelligence, and then skillfully build a platform and set up a series of questions with variable levels, so that students can train moderately in imitation, actively migrate in analogy, expand and sublimate in innovation, and construct knowledge in spiral rise. Compared with the "addition" teaching mode of "directly talking about the topic", this kind of "multiplication" exercise teaching not only saves a lot of time and repetitive work, but also makes the connections and differences of a series of problems appear together, sublimates understanding in comparison, and prints countless connections in students' minds. The students trained in this way are good at analogy, migration, understanding, clarity and easy learning after entering high school, which can effectively avoid the embarrassment of "understanding and doing wrong".
In the example of this lesson, the series of variant problem sets of fragments 2 and 3 not only guide students to internalize their understanding and consciously establish the optimal scheme in exchange and exhibition, but also strengthen the training of the core idea of this lesson-using the symmetry of quadratic function to solve problems in the gradual change from special to general.
5. Ask more questions and worship less.
In teaching practice, we sometimes have to face such an embarrassing scene: the top students in the class taught by controversial teachers are of high quality and quantity, but the "excellent" teachers recognized and worshipped by students are dwarfed and obviously at a disadvantage. This is closely related to students' "questioning" spirit. If you don't trust the former, you should question it more and try to verify each question yourself. I am willing to do the latter, but I can't do it. I fully agree with what the teacher said and did, and then I put it on the shelf and ignored it.
Therefore, in junior high school mathematics teaching, while giving up the leading role to students, we should also "deliberately" play hard to get, or naive thinking, or set traps, or stay "half awake and half drunk", or keep suspense when the plot is ups and downs, just like TV series, cultivate students' problem consciousness and let students have doubts and doubts. Once students' enthusiasm for asking questions is stimulated, their learning initiative will be like a "chain reaction", their potential learning ability will bloom, the classroom will be full of vitality, and students' thirst for knowledge will remain vigorous. From in-class to out-of-class, students will comprehensively use what they have learned in a wide range and from many angles, and even have a whim.
In the third segment of this class, the teacher gave the pointer to the students, and the students took the initiative to reveal secrets, ask questions and solve doubts, which achieved good teaching results, but I am afraid there are more opportunities to solve doubts. In the long run, after three years of high school, the way the class asks questions will definitely be directly proportional to the academic performance.