Reflections on the teaching of acute trigonometric function in junior high school mathematics (1) What else should be taught about trigonometry? How to teach? This immediately became a big problem for some teachers.
In view of this, I think it is necessary to re-examine this part of the knowledge system and clarify new teaching ideas, so as to truly implement the opinions of this adjustment and realize? Three benefits? It is beneficial to reduce students' excessive academic burden, deepen the curriculum reform of ordinary high schools and stabilize the teaching order of ordinary high schools.
First of all, is it? Triangle? Or? Function?
It should be said that trigonometric function is composed of? Triangle? And then what? Function? It consists of two parts of knowledge. Triangle is a derivative of geometry, which originated from Hipparchus in ancient Greece and passed through Ptolemy and Litex Lovech Pcf. In Euler, it finally became a well-structured and thriving classical mathematics discipline. For a long time in history, only trigonometry prevailed in the world, but there was no trigonometry. Trigonometric function? Name.
? Trigonometric function? The concept appeared naturally after the functional concept, but it was only more than 300 years ago in terms of time. However, once this concept was put forward, it immediately changed the face of trigonometry greatly. Especially after Lobachevsky's pioneering work. Thus, trigonometric function can be completely independent of triangle and become a branch of analysis, in which the angle is not limited to the positive angle, but an arbitrary real number. Some scholars even think it can be called angular function, which is quite insightful.
Therefore, trigonometry as a discipline no longer exists independently. The current middle school textbooks also cancel the original pattern of algebra, triangle and geometry, and incorporate triangles into algebra. What does this in itself mean? Function? Are you online? Triangle? The proportion that should be accounted for.
Judging from the historical evolution of algebra, in a long historical period, Formulas and equations? It has always been its core content, and the textbooks at that time were all around it. Therefore, the fractional deformation, radical deformation, exponential deformation and logarithmic deformation in the book can be described as endless and ubiquitous. This is determined by the level of mathematical cognition at that time. Now, function has replaced formulas and equations as the core content of algebra, and people pay more attention to the cognitive value and application value of function thought, rather than operational skills and deformation routines. 1963 promulgated by the mathematics syllabus puts forward three abilities of mathematics, which should be emphasized first? Formal calculus ability? What is highlighted in the outline of 1990? Logical thinking ability? . The images, properties and applications of power function, exponential function and logarithmic function are fully expounded in the current senior high school algebra textbooks, but the deformation of these three algebraic expressions is understated.
So, the trigonometric function part should be focused? Images and properties of functions? There is no doubt that this is also an internationally recognized view (discussed below).
There is also a separate chapter on trigonometric functions in high school algebra. What are the images and properties of trigonometric functions? This is consistent with the development trend of mathematics. But if trigonometric function is mentioned, what is reflected in the minds of most teachers and students Countless formulas, countless transformations? , and trigonometric functions? Images and attributes? But second. This, is it on it? Strength, fingers, right? The function contrast is very big, and I am afraid it is also very different from the original intention of the editor. Of course, the reason is related to the multiple formulas of the triangle itself, and the interference effect of Formula 8, which is reciprocal to the product, is particularly obvious. Formulas are similar in form, and it is not easy to remember, especially to master deformation, which is the same difficulty for teachers and students in teaching and learning. Therefore, it is inevitable to repeat memory and exercise in the sea.
After the adjustment, the requirements of this part are reduced and the number of questions is greatly reduced. First and third? This can achieve two advantages. But how to understand the other (conducive to deepening the curriculum reform)? Release? Function? As a key word, the point? Images and attributes? In fact, it should be the right choice, with light burden and small obstacles, which is more convenient for us to turn our attention to the image and essence of function. What is this? Three benefits? The foundation of realization.
Second, foreign views and enlightenment
Let's look at the views of the United States and Germany:
The United States does not have a unified national textbook and exam instructions, but only a curriculum standard. In the curriculum standards, they put forward the following requirements for trigonometric functions:
Can use the knowledge of trigonometry to solve triangles; Can use sine and cosine functions to study periodic phenomena in objective reality; Master trigonometric function images; Able to solve trigonometric function equation; Will prove the basic and simple trigonometric identities; Understand the relationship between trigonometric functions and polar coordinates and complex numbers.
Reflections on the teaching of acute trigonometric function in mathematics in grade three (2) The progress of mathematics teaching in grade nine is relatively tight, and it is estimated that the new course will be finished before the Spring Festival. However, due to various reasons, after the Spring Festival, there are two chapters in our ninth grade mathematics, namely, acute trigonometric function and projection and view. Therefore, we should improve the teaching speed to speed up the progress, end the new course as soon as possible and enter the general review. Under such circumstances, it is even more necessary to rationally integrate teaching materials, apply the principle of students' speed and slowness, take root, and let students learn by themselves with derivative knowledge. In recent years, the theory in this area is actually quite clear, but in practice, it will always be very different.
For example, in the teaching of acute trigonometric function, I want to integrate the contents of the original two weeks 1 1 class into six classes. The whole idea of preparing lessons is as follows: in the first class, let students fully understand what is relative, adjacent, oblique and some proportion conversion; In the second class, I was fully familiar with the correspondence between various corners of sine, cosine and tangent functions. In the third class, after fully explaining the angular relationship, let the students explore how to use the angular relationship between various functions to calculate the right triangle. Lesson 4: Special angle and its calculation; The fifth category: application. The sixth class: test. I thought at that time, if I understood all these questions, the study of this chapter would be basically no problem. But the test results came back. What happened? Reluctant to gamble? Many students are not familiar with what cosine is, let alone its application. It happened that I took part in the evaluation of this test and participated in an open class of a science group. At that time, the headmaster and colleagues found this problem and suggested some improvement methods to me, which prompted me to seriously reflect on this chapter? Ideal arrangement? The essence of this chapter is the angular correspondence of sine, cosine and tangent functions and their clever use. Recalling this chapter, I also want students to feel the class. Playing in the water Yes, but it didn't work? Take root in the roots? Well, basically, I'm still trying to catch up with my classes and don't want to spend enough time with my students. Shallow pool swimming? Yes