I. Origin
This is an exercise class, which introduces the positional relationship between straight line and hyperbola. I showed the following examples:
Known hyperbolic x? 24-y？ 22= 1, find out whether there is a straight line l, so that the midpoint of the hyperbolic chord is N( 1, 12).
The following is part of my actual teaching process:
[Teacher] Just now, we established the linear equation with the point oblique method, connected the straight line L with a hyperbola, and expressed the point N( 1, 12) with the relationship between roots and coefficients, and then got the slope of the straight line L. Of course, we found that the straight line L did not meet the meaning of the question. Undoubtedly, the positional relationship between a straight line and a conic curve can be solved simultaneously. So can we use the slip method?
[student 1] suppose that a straight line l exists, and let the two intersections of the straight line l and the hyperbola be A(x? 1，y？ 1)，B(x？ 2，y？ 2), respectively, into hyperbolic equations. Put two x's? 1? 24-y？ 1? 22= 1，x？ 2? 24-y？ 2? 22= 1 subtraction, you can get:
14(x？ 1-x？ 2)(x？ 1+x？ 2)- 12(y？ 1-y？ 2)(y？ 1+y？ 2)=0, from the meaning of the question, x? 1+x？ 2=2，y？ 1+y？ 2= 1。
Instead of y? 1-y？ 2x？ 1-x？ 2= 1, so the slope of the straight line L is 1, and the equation of the straight line L is y = x- 12.
Student 2: No, I just found that the straight line Y = X- 12 does not intersect with the hyperbola when I used "joint legislation", so I have to check here, and the solved straight line Y = X- 12 should be abandoned.
[Teacher's guidance] What kind of questions can be slipped?
Student 3 When it is related to the midpoint and slope of the chord at the intersection of a straight line and a hyperbola, the point difference method can be used.
Of course, the linear equation solved is to test whether it meets the meaning of the question.
Second, doubts-inquiry
Just when I thought the problem had been solved satisfactorily, some students raised questions.
Student 4: Why does the straight line L whose equation is Y = X- 12 have no intersection with hyperbola?
I was shocked to hear that. This is a problem that I didn't think of in the process of preparing lessons! But I still want to explore with my students.
[Teacher] Then let's take a look at the whole process of answering questions. What might be wrong with this method?
The students began to explore, and soon. ...
Student 5: The problem lies in business. As long as the hyperbolic equation satisfies x? 24-y？ 22=m(m≠0), and then after substituting the point, put x? 1? 24-y？ 1? 22=m and x? 2? 24-y？ 2? 22=m can get 14(x? 1-x？ 2)(x？ 1+x？ 2)- 12(y？ 1-y？ 2)(y？ 1+y？ 2)=0。
So change the hyperbola in the question to x? 24-y？ 22=m(m≠0), that is, the linear L equation can be calculated as y = x- 12.
[teacher] in all hyperbolas x? 24-y？ 22=m(m≠0), some have two intersections with the straight line L, and some have only one intersection with the straight line L or no intersection. So, when does a hyperbola intersect a straight line?
[Student 6] Substitute the equation Y = X- 12 of the straight line L into x? 24-y？ 22=m(m≠0), calculation? Δ? = 2- 16m, as long as? Δ? & gt0, that is, m ∈ (-
Symboleb @, 0)∩(0, 1 16), the midpoints of the chords intersecting with the straight line y = x- 12 are all N( 1, 12). Among them, when m ∈ (-
When symphony @, 0), the intersection of hyperbola falls on the Y axis, the straight line Y = X- 12 intersects hyperbola and the midpoint of the intersecting chord is n (1,12); When [JP3]m∈(0, 1 16), the intersection of hyperbola falls on the x axis, and the straight line Y = X- 12 intersects hyperbola at the right branch, and the midpoint of the intersecting chord is N( 1,/kloc-0).
Third, some reflections after class.
1. The choice of mathematical problems should be enlightening, exploratory and open.
Bolivia said: "The good problem is similar to mushrooms, most of which grow in piles. After finding one, you have to look around, and there are probably several nearby. " Use a meaningful but not too complicated topic to help students explore all aspects of the problem, so that through this topic, it is like introducing students into a complete theoretical field through a portal. In short, the design of problems should pay attention to strategies, try our best to ignite students' thinking sparks, stimulate their desire for knowledge, consciously provide them with bridges and ladders to solve problems, and guide them to gradually master brand-new knowledge and abilities.
2. Teachers should not only guide students to explore their own whimsy, but also make students' whimsy "dazzling".
Mr. Tao Xingzhi, a famous educator, said: "everywhere is a place of creation, and it will always be an era of creation. Everyone is a creator." In the teaching process, teachers should believe in students' ability and let them talk and discuss. It can be said that in the discussion, problems disappear and students' confidence doubles. Teachers should not be the presenters of knowledge and the symbols of knowledge authority, but should listen to students' thinking process patiently in the spirit of learning from each other's strengths, because only by seeking differences can there be innovation, and students' "whimsy" is likely to be a "wonderful solution", which may contain innovative thinking and sparks of wisdom. Students' unique thinking should be properly evaluated and encouraged. Through proper evaluation, we can awaken students' innovative potential, stimulate students' innovative desire, and finally realize students' self-worth.
(Author: Jiangsu Yunhe Middle School)