In the reflection on the mid-term examination of mathematics in the first semester of 2020-202 1 school year, the author deeply thought about the problem of 18.
As shown in the figure, it is known that ∠ EOF = 90, in ABC, AC = BC = 10, AB = 12, and point A and point B move on OE and OF respectively, and the shape and size of ABC always remain unchanged. In the process of movement, the maximum distance from point C to point O is. ? This question is a 4-point fill-in-the-blank question, and the scoring rate is extremely low. There are 795 second-year reference students in the whole school, 18 classes, with an average score of 1.05, each of whom lost nearly 3 points on this topic. See the following table for scores:
Judging from the results of each class, the lowest score is 0.43, the highest score is 2.77, and the height difference is 2.34. Why is there such an unusual result? It is understood that the teachers of 15 and 16 briefly introduced the solutions to similar problems at the request of students before the exam. Themes and explanations are as follows:
As shown in the figure ∠MON=90? It is known that ABC, AC=BC=5, AB=6, and the vertices A and B of ABC are ON the edges OM and ON respectively. When B moves on the edge ON, A moves on the edge OM, and the shape of the triangle ABC remains unchanged. In the process of movement, the maximum distance from point C to point O is _ _.
[Pythagorean theorem; The nature of isosceles triangle; The midline on the hypotenuse of a right triangle; Triangular trilateral relationship]
Take the midpoint d of AB. Connect the CD. According to the trilateral relationship of the triangle, OC is less than or equal to OD+DC. Only when the three lines of O, D, C*** and OC take the maximum value is OD+CD. According to D as the midpoint of AB, OD=3 is obtained. According to the combination of three lines, CD is perpendicular to AB. In RtBCD, according to Pythagorean theorem, the length of CD is 4, and then DC is obtained.
As shown in the figure, take the midpoint d of AB and connect CD. AC = BC = 5，AB=6。 ∵ Point D is the midpoint of AB, ∴BD=3, from Pythagorean theorem, CD = 4；; Connect OD, have OC? OD+DC, when O, D.C*** is connected, OC has a maximum value, which is OD+CD, and ∵AOB is a right triangle, and D is the midpoint of the hypotenuse of ∴od=3 ab, that is, OC=7. As shown in the figure below:
Push grinding principle
As can be seen from the picture of the grinding mill, the length of the grinding rod is unchanged, one end is fixed at the edge D of the grinding disc, and the other end is inserted at the midpoint C of the grinding rod (handrail). The grinding rod is horizontally suspended in the air with two ropes, and the length of the ropes is constantly adjusted to make it (horizontally placed grinding rod) on the same plane as point B as possible. In the process of grinding, the human body leans forward from time to time, leans back from time to time, and goes on and on. Abstract the grinding principle into a mathematical problem is:?
As shown in the figure, in O, O stands for grinding disc, and its center point O is a fixed point (grinding center) with constant radius, and CD is a grinding rod with constant length. The position of moving point C changes with the movement of moving point D on O. When point D moves to coincide with the position of point Q, points O, D and C are * * lines. At this time, ODC does not exist and point C arrives. When point D moves to coincide with the position of point P, points D, O and C are * * * lines. At this time, ODC does not exist, and the distance from point C to center O is the smallest, and the minimum value is the value of CD-OD. Using this principle, we can solve a kind of mathematical and geometric maximum problem in the senior high school entrance examination. We might as well call this principle "the grinding principle". .
In solving the above problems, the teacher first pointed out that the midpoint D of AB should be taken to connect OD and CD, then the lengths of OD and CD should be calculated respectively, and finally the value of OD+CD, that is, the minimum value of OC, was calculated, and the problem was solved. The question is why this question should take the midpoint d of AB. The key point is that students don't know that we need to take the midpoint D of AB. Therefore, the scoring rate of this question is extremely low, and it is not surprising that everyone drops nearly 3 points on this question on average. If we can emphasize to students that when we see the problem of finding the minimum value of OC, we should contact the "grinding principle" and students will have the direction to solve the problem: we must find such a D point! And this point is basically related to the midpoint of the hypotenuse of a right triangle.
Mathematics comes from life, and life serves life. The grinding principle is the best example. The author tells the "grinding" in life from concrete examples, summarizes the general methods to solve problems, and then uses it to help us solve practical problems. Thus, the vivid mathematical model "grinding principle" is organically combined with boring mathematical problems, so that students can have a goal direction when solving such problems. This kind of mathematics teaching is really valuable mathematics with flesh and blood.
Zhang Tingwei is a famous math teacher in Shu Ren Middle School in Xuzhou City, Jiangsu Province. He combined traditional teaching with wisdom teaching, formed the mode of "teaching with learning", refined the "three principles" of mathematical interpretation, and exchanged experiences at the national seminar on wisdom teaching, which was deeply affirmed by the delegates and experts.
(office editor)