Abstract: Function knowledge is very important in junior middle school mathematics. To do this part of teaching well, we must understand the basic concepts, clarify the knowledge structure and establish? Sports changes? The idea of combining numbers with shapes.

Keywords: the combination of numbers and shapes in junior high school mathematics function teaching

The introduction of the concepts of variables and functions in junior high school mathematics marks the progress of mathematics from constant mathematics to variable mathematics.

Although the content of junior middle school function only tells some basic and preliminary knowledge of function, the mathematical ideas and methods contained in it are very beneficial to cultivate students' ability to observe, study and solve problems.

Moreover, the concept of function is the core part of high school algebra. Learning the knowledge of junior high school functions well can lay a certain foundation for learning various elementary functions in senior high school mathematics.

Therefore, the basic function of junior high school function concept is obvious.

In teaching, students should be guided to correctly understand the concept of function from four aspects, and then master the characteristics and properties of function.

First, correctly understand the three groups of relationships and systematically grasp the concept of function.

The definition of point coordinates and the one-to-one correspondence between points and coordinates; The changing process of function definition and the corresponding relationship between independent variables and functions; The value of independent variables in the definition of function images.

Function value? Ordered numbers, right? The coordinates of the point? Point? Image, strengthening the understanding of these three groups of relations, is conducive to combining the analytical formula of function, the coordinates of points and the image of function, and establishing a relatively complete concept of function.

Second, clarify the knowledge structure and build a knowledge system.

With such a knowledge structure diagram, we can organically combine the plane rectangular coordinate system, points, images and analytical expressions to find the relationship between them and the transformation mode of the problem.

Third, establish the viewpoint of movement change.

The core meaning of the concept of function is to reflect the dependence between two variables in a certain change process, that is, the change of one quantity changes with the change of the other.

This establishes a dynamic relationship between the concepts of static numbers.

In the teaching process, students should be guided to understand this variable relationship by finding and discovering examples around them.

For example, the height of the growing period changes with age; The temperature in a day changes with time; The income of the factory increases with the increase of output; There are countless solutions to the binary linear equation. In the equation 3x-2y= 1, when the value of X changes, the value of Y changes with the change of X..

While expounding this movement relationship, we should also use formulas, tables and charts to describe it, so as to deepen students' intuitive understanding of this abstract movement relationship and help students gradually establish a kind of? Sports changes? The point of view.

Fourth, cultivate the idea of combining numbers with shapes.

Mathematics teaching process should reflect two lines of light and dark: one is the bright line, that is, the teaching of mathematics knowledge content; The other is dark line, that is, the formation of mathematical thinking method.

Because mathematical thinking method is not only the basic knowledge of mathematics, but also the bridge from knowledge to ability, using mathematical thinking well is to develop mathematical ability.

Therefore, in teaching, teachers should pay attention to cultivating students' infiltration, generalization and summary of mathematical thinking methods and improve students' application ability.

The thinking method of combining numbers and shapes is an important thinking method in junior high school mathematics.

What is the thinking method of combining numbers with shapes? As we know, mathematics is a science that studies the quantitative relationship and spatial form in the real world, and number and shape are two basic concepts in the mathematical knowledge system. We organically combine numbers describing quantitative relations with concrete and intuitive graphics, and organically combine abstract thinking with image thinking. According to the need of discussing the problem, the comparison of quantitative relationship can be transformed into the discussion of image properties or their positional relationship, or the undetermined relationship between graphics can be transformed into the quantitative calculation of related factors, that is, the flexible transformation and interaction between numbers and shapes, so as to explore the solution to the problem.

In this part of the function, there are rich mathematical ideas, such as coordinate ideas and the combination of numbers and shapes, the most important of which is the combination of numbers and shapes.

So how to infiltrate and apply the thinking method of combining numbers and shapes in the teaching process of functions is particularly important.

For example, a linear function is a straight line, and the coordinates of points on this straight line satisfy the analytical formula no matter how they change.

A straight line is made up of points and can be described by numbers.

On the contrary, straight lines reflect the changing characteristics of numbers.

A function can be represented by a graph, and some properties and characteristics of the function can be analyzed intuitively with the help of this graph, which provides great help for the research and application of mathematics. Teachers will get twice the result with half the effort if they pay attention to the infiltration of the thinking method of combining numbers and shapes in teaching.

The common styles in junior high school mathematics teaching are: (1) the corresponding relationship between numbers and points on the number axis; (2) correspondence between function and image; (3) The correspondence between curve and equation; (4) The concept based on set elements and geometric conditions; (5) The structure of a given equation or algebraic expression has obvious geometric significance.

Of course, the above content is only my own experience in teaching practice. In fact, the content and requirements of junior high school functions are extremely rich. Cultivating students' thinking ability and being able to use knowledge flexibly is the ultimate goal of our study. When discussing social problems, economic problems, interdisciplinary synthesis and other issues, more and more mathematical ideas and methods are applied, among which the content of functions occupies a very important position.

Therefore, in the process of teaching and learning, we must carefully study the textbooks and dig deep into the ideas, methods and viewpoints contained in the textbooks in order to improve students' thinking ability, application ability and cognitive level.

Function teaching in junior high school II

Mathematical thinking method is the law and essence of mathematics. If students master mathematical thinking methods, they can acquire knowledge faster and understand knowledge more thoroughly.

Function teaching in junior middle school should teach students to master the thinking method of learning function.

This paper only makes a preliminary exploration of function teaching in junior high school.

Keyword function teaching

First of all, understand the function thought and guide the teaching direction.

Function describes the dependence of quantity in nature and reflects the relationship and law of one thing changing with another. The thinking method of function is to extract the mathematical characteristics of the problem, put forward the mathematical object with the viewpoint of contact change, abstract its mathematical characteristics, establish the function relationship, and study and solve the problem by using the properties of the function.

Although the content is not much, the thought of function has been embodied and still occupies an important position.

Second, clear the concept of junior high school function and systematically master the knowledge of elementary function.

1, understand the logic of the concept.

Mathematical concepts can be divided into two important aspects: first, the' quality' of concepts, that is, the connotation of concepts (the essential attributes of concepts); Second, the' quantity' of the concept is the extension of the concept (the sum of all the objects of the concept). The extension of the concept can be divided into different sizes. The concept with large extension is called generic concept, and the concept with small extension is called generic concept. The essential difference between a generic concept and other generic concepts is also called generic difference. If you want to make a final decision on a concept, you should first point out what is the closest concept to the defined concept, and then point out the difference of the defined concept, that is, the concept definition = A.

2. Clarify the level of concepts.

General concepts are obtained through abstract generalization of experimental phenomena or analysis of specific things. This is a process of formation. Many concepts in middle school are developed into a series of definitions and kilometers from several original concepts and axioms through certain reasoning, and each new concept is expressed or overthrown by the old concept.

3. Grasp the abstraction of concepts.

Many original concepts in junior high school mathematics are formed by the perception of specific numbers and shapes, and then abstractly summarized from the representation.

Concept is the product of people's abstraction of perceptual materials, and perceptual knowledge is the basis of forming concept.

If students have no perceptual knowledge or incomplete perceptual knowledge, they should use materials, models, multimedia courseware or images to teach more intuitively, so that students can get perceptual knowledge from them.

Third, draw the image of elementary function and understand the essence of elementary function.

Mr. Hua, a famous mathematician, said: "Missing numbers is less intuitive, while missing numbers is difficult to be nuanced."

Therefore, in order to draw the image of elementary function and understand its properties, we must first understand the idea of "combination of numbers and shapes".

The problem of large numbers in mathematics implies the information of shapes, and the characteristics of figures also reflect the relationship between numbers.

It is necessary to abstract the complex quantitative relationship and reveal it intuitively through the image of form, so as to achieve the goal of integrating form and number.

Fourthly, to cultivate students' interest in learning functions by using the connection and practice between functions and other disciplines.

The function is defined as follows: "If two variables X and Y are set in a certain change process, and Y has a unique fixed value corresponding to each fixed value of X within a certain range, then Y is called a function of X, X is an independent variable and Y is a dependent variable".

As shown in figure 1 (1), in the rectangular ABCD, AB= 10cm and BC=8cm.

Point p leaves from point a and follows route a? b？ c？ D movement, stop at point d; Point q starts from point d and follows d? c？ b？ Route a moves and stops at point a.

If P and Q are started at the same time, the speed of point P is 1cm/s and the speed of point Q is 2 cm/s. ..

In one second, the speeds of P and Q change at the same time. The speed of point P becomes b cm/s, and the speed of point Q becomes d cm/s. ..

Figure 1 The second figure is the functional relationship between the area of △APD S 1 (square centimeter) and x (second) after x seconds from point P.

Figure 1 The third figure is the functional relationship between the area S2 (square centimeter) of △AQD and x (second) after x seconds from the Q point.

2. Function and market economy

Example 2: A chemical raw material sales company purchases 7,000 kilograms of a chemical raw material, and the purchase price is 30 yuan per kilogram.

The price department stipulates that its sales unit price shall not be higher than that of 70 yuan per kilogram, nor lower than that of 30 yuan.

The market survey found that the unit price is set at 60 kilograms per day in 70 yuan; The unit price is low 1 yuan each time, and it sells 2 kilograms more every day.

During the sales process, other expenses will be paid every day, 500 yuan (if the number of days is less than one day, it will be counted as a whole day).

Suppose the sales unit price is X yuan and the daily average profit is Y yuan.

Vertex coordinates are (65, 1950).

Schematic diagram of quadratic function (as shown in Figure 2).

According to the sketch, when the unit price is set to 65 yuan, the daily average profit is the largest, which is 1950 yuan.

(3) When the average daily profit is the highest, the unit price is 65 yuan, and the average daily sales is 60+2? (70-65) = 70kg, so the total profit is 1950? (7000? 70)= 195000 yuan

When the sales unit price is the highest, the unit price is 70 yuan, and the average daily sales volume is 60kg. It takes 700 yuan to sell all these chemical raw materials. 60? 1 17 days.

So the total profit is (70-30)? 7000- 1 17? 500=22 1500 yuan

∵22 1500 & gt； 195000 and 221500-195000 = 26500.

? When the sales unit price is the highest, the total profit is the highest, and the extra profit is 26500.

It can be seen that functions are widely used, closely linked with other disciplines, and are important tools to solve practical problems, so they can improve and cultivate students' interest in learning elementary functions.

Nowadays, with the rapid development of science and technology and the rapid increase of scientific knowledge in the world, students have many complicated problems to understand, discuss, analyze and solve in their future work and life and further study. We should give students the thinking method of function as the golden key, and let them use this golden key to open the treasure house of knowledge and meet the challenges of new life!

Function teaching in middle school 3

From the development process of mathematics itself, the introduction of the concepts of variables and functions marks the progress of mathematics from constant mathematics to variable mathematics. Although the content of junior middle school function only tells some basic and preliminary knowledge of function, the mathematical ideas and methods contained in it are very beneficial to cultivate students' ability to observe, study and solve problems.

Keywords learning interest situational teaching

Function is an important mathematical knowledge in junior high school mathematics, and the quality of function learning has a far-reaching impact on students' continuous learning. In particular, the new curriculum standard puts forward research-based learning, pays more attention to the cultivation of students' ability to read pictures, and tries to solve problems with the idea of combining numbers with shapes.

Based on many years of mathematics teaching in middle schools, the author talks about the following thoughts on how to do well the function teaching in middle schools.

First, clarify the importance of learning function and cultivate students' interest in learning function.

The concept of function plays a lateral role in the main contents of junior middle school mathematics, such as formulas, equations and inequalities. Essentially, algebraic expressions can be regarded as analytic expressions or values of functions; The two algebraic expressions a and b are equal in value to the function y=A-B is equal to zero; The root of the equation can be regarded as the abscissa of the intersection of the function image and the X axis. In the proof of inequality, the nature of function is often a powerful tool.

Because function is widely used, and the formation and development of function concept is a leap from constant to variable in middle school mathematics, it is undoubtedly helpful to understand and master the thinking method of function to realize this leap.

The function of our junior middle school is relatively simple, which belongs to the function enlightenment, but it is the main content of high school mathematics and even the whole mathematics system. Therefore, the junior high school stage is the key stage of the formation of function concept and function thought. The success or failure of this stage of teaching is directly related to students' mathematics learning in high schools and universities, and even their lifelong mathematics literacy.

Let students fully realize the importance of function, which is helpful to improve their interest in learning function.

Second, situational teaching

Teachers can put forward mathematical knowledge points in the form of questions, stimulate students' desire to learn, deepen their thinking about knowledge points in the process of thinking, and create situations to provide them with thinking space, so that their thinking can transition from image to abstraction and complete the transformation of thinking. In classroom teaching, many problems depend on students' imagination, but it is impossible for students to feel every problem outdoors, which requires us to strengthen students' abstract thinking ability, especially when learning functions.

The ultimate goal of learning function knowledge is to be able to apply it in real life. Therefore, when teaching functions, teachers use materials in specific situations as materials to inspire students' thinking, and strengthen students' understanding of knowledge points through mutual communication, cooperative learning and independent thinking.

Students can better grasp the understanding of the problem in the problem situation. In the problem situation, teachers should give some guidance and help. Teachers should follow the step-by-step and step-by-step understanding method to create problem situations for students and create learning opportunities. By inviting students to swim in problem situations, students can bathe in math activities. Problem situation is an effective method to strengthen mathematical understanding and problem solving.

Third, adhere to the concept of mutual connection and sports development for teaching.

Function shows the interdependence between two variables, one variable will change with the change of the other variable, and they are the order of mutual restriction and development, and there is a moving connection between the concepts of seemingly static numbers.

In junior middle school function teaching, teachers should guide students to learn the basic knowledge of function and the methods to solve problems, cultivate students to establish mathematical concepts of mutual connection and sports development, and master the basic essentials of function knowledge in dynamic thinking mode.

The interaction between two variables is not easy for students who are new to function knowledge to understand.

Junior high school function teacher can according to? One quantity changes with another? This relationship allows students to combine familiar mathematical knowledge with examples of daily life practice, such as? The fuel consumption of a car varies with the distance traveled? , or? Does the area of a circle vary with radius? Wait a minute.

In this way, students can understand the definitions of independent variables and variables more quickly, and can exercise their ability to analyze and solve problems in an active thinking environment.

The relationship between variables in the function has a comprehensive relationship with many fields in the mathematical knowledge system, such as finding the distance? Distance = speed * time? And so on, reflecting the importance of function.

Learning function knowledge actually opens up more perspectives in the field of mathematics.

In addition, function is closely related to other disciplines and is an important tool to solve practical problems.

Junior high school math teachers can use the extensive connection of functions to stimulate students' enthusiasm for learning in the process of recruiting and introducing, so as to achieve real teaching results.

Fourth, pay attention to the application of analogy in interpretation.

When explaining the image of a function, we usually derive it from a special case.

For example, draw the image of the following function in the same rectangular coordinate system: (1) y = 2x+3 (2) y = 2x+5 (3) y = 2x-3; (4)y=-2x+3(5)y=-2x-3

Then the students draw a conclusion that the image of the function is a straight line, and ask them to draw from the above image: when (1) k >; 0，b & gt0 ;

(2)k & gt； 0，b & lt0; (3)k & lt； 0，b & gt0; (4)k & lt； 0, b<0, the quadrant and monotonicity of the function image are summarized by the teacher, and the students understand and remember.

This program is so general that it is difficult for students to remember it.

Ask students to recall the proportional function (1) y = 2x; (2) The image and properties of 2)y =-2x, and then draw the image of the above function, and get the image and properties of the primary function by analogy.

Show students the translation changes of the proportional function image, that is, get a function image, which can prevent students from separating them, grasp their uniqueness and distinguish the particularity of the proportional function.

Cultivating students' knowledge transfer ability through analogy.

Fifth, strengthen the communication between disciplines and enhance students' awareness of using mathematics.

One of the directions of current educational reform is to strengthen the comprehensive application of knowledge in various disciplines.

As a basic discipline, mathematics not only serves other disciplines, but also helps to understand the application of mathematics and cultivate students' comprehensive ability if we can combine the characteristics of other disciplines and explain its basic principles with the knowledge of other disciplines.

Example 3: The original length of the spring is 15cm. As we all know, the length of the spring has a linear relationship with the suspension mass within 20 kg.

When the hanging weight is 4 kg, the length of the spring is 17cm. When the spring length is 22cm, what is the hanging weight?

Analysis: Based on the known condition that the spring length is linearly related to the suspension weight, the functional relationship is obtained by the undetermined coefficient method.

The solution of this problem can be obtained by calculation.

Solution: Let the hanging weight x(kg)(0? x？ 20), the length of the spring is y(cm), which can be set according to the meaning of the question, y=kx+b (k? 0) According to the condition: when x=0, y= 15, that is, b= 15.

When x=4, y= 17, that is, 4k+ 15= 17, so K=

Therefore, the resolution function is: y= x+ 15 (0? x？ 20)

So when y=22, x+ 15=22, x= 14.

Answer: When the spring length is 22cm, the hanging weight is 14 kg.

For physical problems, we must list the functional relationships according to physical concepts and physical knowledge, turn them into mathematical problems, and then use mathematical methods to operate them, as well as other disciplines.

In a word, how well the function is learned in middle school will directly affect students' interest in mathematics learning and the quality of their grades in the future. Therefore, the majority of middle school mathematics teachers shoulder the key responsibility and must attract our great attention.

The above points are my humble opinions, and I hope to give some help to my peers. Please correct me.

refer to

[1] sheets. On Function Teaching in Junior High School [J]. Learning, 2009( 15).

[2] Xu Deben. What should be grasped in junior middle school function teaching? Four ones? [J]。 Middle school mathematics teaching reference .2008, (18).

[3] Wang; Junior high school students' learning function difficulties and teaching strategies [J]: success (education); 20 1 1 18