Functional relationship is the abstraction of the relationship between quantity and quantity. When it comes to the relationship of quantity, we should use the concept of function to describe and depict it, and study the relationship of quantity in objective reality through it. Therefore, whether it is employment or further education, we must learn some functional concepts.

The algebra textbook in senior high school is centered on function, which is abstract and difficult to learn, so junior high school should talk about function to prepare for senior high school.

As far as junior high school algebra itself is concerned, it solves triangles, quadratic inequalities and so on. It is also inseparable from the related concepts of functions. In physical chemistry, such as uniform motion, Boyle's law, projectile motion, free fall, etc., there must also be corresponding functions as the basis.

Therefore, the learning function of junior high school is quite necessary.

Second, the characteristics of junior high school function teaching

First of all, from the whole middle school stage, function teaching can be roughly divided into the following three stages:

First, the stage of perceptual knowledge.

This stage is characterized by the accumulation of materials, which basically belongs to this stage before the concept of function is formally introduced.

The basic contents of this stage of teaching are as follows:

(1) Through various mathematical operations, let students observe the relationship between the operation results and the components of the operation, such as the relationship between sum, addend and quotient, dividend and divisor.

(2) Through the study of algebraic expressions and equations, students can further understand how to express general quantitative relations in words; How to express the relationship between quantity and quantity with algebraic expressions?

(3) Through the development of the concept of number, students' initial ideas about the concept of "set" can be accumulated. For example, when talking about the allowable value of the root sign, students can be guided to pay attention to non-negative sets. The textbook consciously permeates some fixed ideas, which is very helpful for the discussion of function concepts in the future.

(4) Accumulate the preliminary ideas of corresponding concepts through the teaching of number axes and coordinates.

Second, the stage of rational understanding.

This stage is the main stage of function teaching. It is divided into two small periods. The first cycle is the "function and its image" in junior high school; The second cycle is to talk about trigonometric functions and their images from the set in high school. The teaching task at this stage is to form the general concept of function correctly, deeply understand the relationship between functions, master the method of drawing simple function images and discussing their properties, learn to apply the properties of functions to solve some simple practical problems, and push students' cognitive level and thinking level forward.

Third, deepen the development stage.

The main task of this stage is to understand the changing trend of the function and master the limit method-infinite precision method through it; Using the tool of calculus, this paper makes a further study on the increase, decrease and extreme value of functions, and points out the limitations of studying functions with elementary methods.

These three stages are connected with each other, which shows that the function teaching in junior high school plays a connecting role, and its learning quality will directly affect the later learning.

Secondly, the teaching of junior middle school function is descriptive, whether it is the teaching of function concept or the teaching of function nature. Therefore, accuracy and popularity are its teaching characteristics. Although it is descriptive, the explanation should be accurate, not misleading to students, and easy for students to accept. Therefore, we should give more examples and use more intuitive means such as graphics and tables.

Thirdly, the concept of function.

Regarding the definition of functions, elements are often mentioned. For example, a function consists of three elements: a definition field, a corresponding rule and a value field. This formulation is unscientific, so it is better not to mention elements, but to focus on the essential characteristics of the concept of function, because elements can not fully reflect the essential characteristics.

The concept of function has two essential characteristics: one is "definition everywhere" and the other is "single-value correspondence" (nouns need not be mentioned to students).

"Definition everywhere" means that in the relationship of r: x → y, if the domain and X are equal, then R is the relationship defined everywhere. That is to say, any element X in X has an element Y in Y corresponding to it. Therefore, the conditions defined by each place are as follows.

In the relationship shown in fig. 39, (1) is defined everywhere, while (2) is not.

Single-valued correspondence means that if R is the relation from set X to set Y, and there is only one y∈Y corresponding to any x∈X, then R is said to be single-valued, that is.

(1) and (2) in Figure 40 are single-valued correspondence, and (3) is not single-valued correspondence.

In the function definition of junior high school algebra, the essence is these two items: "For every certain value of X in a certain range (defined everywhere), Y has unique certainty."

The value of corresponds to it (single value correspondence). "If one of these two items is missing, it will not be its function, so it is much clearer to emphasize essential features than elements.

In addition, students should be prevented from treating all functions as formulas, otherwise the extension of the concept of function will be narrowed. Therefore, when teaching the concept of function, we should give examples of functions that cannot be expressed by formulas.

Fourthly, the teaching of function definition domain.

There are two requirements for the definition domain in middle school textbooks: if the definition domain is given by a formula without specific explanation, it refers to the natural definition domain, even if the formula is meaningful, it is also the range of the independent variable X. The textbook also points out that "when encountering practical problems, determining the range of the independent variable of the function must make the practical problems meaningful", so it should be reflected in teaching.

Finding the definition domain of a function involves solving equations, inequalities, fractions, roots and other knowledge. Therefore, it is necessary to test old and new materials and make appropriate requirements in teaching, but the topics should be the most basic, and don't deliberately make some artificial questions, because this kind of training is of little significance.

Fifthly, the teaching of function images.

Because functions often involve infinite sets, generally speaking, images should be infinitely extended, but this is limited when drawing images, and infinity can only be expressed by finiteness. In this way, on the one hand, limited images are required to reflect the main features of infinite images (such as intersection with axes and peaks). ); On the other hand, it should reflect the infinite trend (such as infinitely close to the X axis, etc.). ), these two points are also the general requirements for drawing function images.

Students should master the following skills when describing function images: setting numbers, calculating (or looking up tables), setting coordinate units, punctuation, adding points and connecting with smooth curves.

There are two situations here:

In one case, we don't know in advance what the drawn image looks like or its nature. At this time, the points should be dense and have both positive and negative values. If the independent variable and the corresponding value are large, the coordinate unit can be set smaller. If there are not enough points at the bend, make up the points appropriately. In short, don't let the image go out of shape.

On the other hand, if the image is known in advance, points can be set according to the characteristics of the image. For example, a proportional function only needs to set a point and then connect it with the origin. For a linear function, you can set two points at will. For the inverse proportional function, if k > 0, just set the point in the first quadrant, and the point in the third quadrant can be obtained from the point with symmetrical origin. K < 0, and only the points in the second quadrant and the points in the fourth quadrant can be symmetrical with the origin.

All these skills should be mastered by students.

Attention should be paid to the function of function images in solving equations and inequalities in teaching.

Sixth, about the teaching of inverse proportional function

The definition, image and nature of inverse proportional function are all difficulties in teaching, which is reflected in the fact that the narrative method is very similar to the direct proportional function, which is easy to misunderstand.

(2) The image of inverse proportional function is a curve, not a straight line (the curve appears for the first time), so it is difficult to cultivate the skills of drawing curve images, such as the curve is two branches, the curve does not intersect with any axis, and it is infinitely close to the X axis and the Y axis.

(3) When teaching monotonicity, "the greater the absolute value of negative value, the smaller it will be", which is often confused by the superficial phenomenon of the image and leads to misunderstanding, thus drawing a wrong conclusion about monotonicity.

All these should be taken seriously.

Seven, about the teaching of quadratic function

Quadratic function is the climax and focus of junior high school word learning function. On the one hand, it is closely related to quadratic equation and quadratic inequality, that is, unifying quadratic equation and quadratic inequality from the perspective of function can organically link them; On the other hand, in the teaching of quadratic function, we have to learn important mathematical ideas, concepts and methods such as "horizontal and vertical axis translation", "formula" and "extreme value", so the textbook of quadratic function has important cultivation.

"The meaning of parameter A", "the equation of symmetry axis", "translation along the axis" and "the meaning of extreme value" are all difficulties in teaching. To overcome these difficulties in teaching, we should proceed from students' reality and adopt concrete and vivid methods for teaching.

The difficulty of the questions about quadratic function should be properly controlled, and the questions should be properly classified. The key point is to cultivate the ability to analyze problems.