1, understand the inverse proportional function, and abstract the analytic function of inverse proportional relationship from practical problems;
2. Draw the image of inverse proportional function, and summarize the properties of inverse proportional function with image analysis;
3. Infiltrate the mathematical thought of combining numbers with shapes and the dialectical materialism thought of universal connection;
4. Experiencing the research and application process of mathematics from practice to practice;
5. Cultivate students' observation ability and the ability to find and solve problems by mathematical methods.
Teaching focus:
Combined with image analysis, the properties of inverse proportional function are summarized.
Teaching difficulties: sketching and drawing images of inverse proportional function.
Teaching tool: ruler
Teaching methods: group cooperation and inquiry.
Teaching process:
1, the concept of inverse proportional function comes from reality.
We learned the inverse relationship in primary school. For example, when the distance s is constant, the time t is inversely proportional to the speed v.
That is vt=S(S is a constant);
When the rectangular area s is constant, the length a is inversely proportional to the width b, that is, ab=S(S is a constant).
From the function point of view, in the process of motion change, there are two variables that can be regarded as independent variables and functions respectively, which are recorded as:
(s is a constant)
(s is a constant)
Generally speaking, a function (k is a constant) is called an inverse proportional function.
For example, when the distance s is constant, the time t is an inverse proportional function of v, and when the rectangular area s is constant, the length a is an inverse proportional function of the width b. 。
In real life, there are many examples of inverse proportional relationship. Students can be organized to discuss. The following examples are for reference only.
2. List and draw the image of the inverse proportional function.
Example 1. Draw an image of the sum of inverse proportional functions.
Solution: list
x
-6
-5
-4
-3
1
2
three
four
five
six
- 1
- 1.2
- 1.5
-2
six
three
2
1.5
1.2
1
1
1.2
1.5
2
-6
-3
-2
- 1.5
- 1.2
1
Note: As it is the first time for students to contact the inverse proportional function, it is impossible to infer its general image. It is best to take more points when taking points, and the positive and negative points can be taken symmetrically and separately.
The image of a general inverse proportional function (k is a constant) consists of two curves, called hyperbola.
3. Observe the image and summarize the properties of the inverse proportional function.
I have learned three basic elementary functions before, and I have a certain foundation. Here, we can learn knowledge according to the level of students or under the guidance of teachers.
Show the images of these two functions and ask the question: What properties can you find of the inverse proportional function from the images? And can be proved from the analysis formula or list. (The following answers are for reference only)
The image of (1) is in the first and third quadrants, which can be extended to K >: 0, that is, k>0. The two branches of hyperbola are in the first and third quadrants respectively. From the analytical formula, we can also draw the conclusion that xy=k, that is, the signs of X and Y are the same, so they are in the first quadrant and the third quadrant.
The discussion is similar.
Seizing the opportunity to expound the unity of number and shape also permeates the mathematical thinking method of combining number and shape, and embodies the research process from special to general.
(2) In the image of the function, in each quadrant, y decreases with the increase of x;
As can be seen from the image, when x changes from left to right, the image shows a downward trend. This trend can also be seen from the list. Rational number division illustrates the same truth. When the divisor is fixed, if the divisor is greater than zero, the larger the divisor, the smaller the quotient. If the divisor is less than zero, the larger the divisor, the smaller the quotient. 0, the image of the function, in each quadrant, y decreases with the increase of x.
You can also infer the nature of the image.
(3) The image of the function does not pass through the origin and does not intersect with the X axis and the Y axis. It can also be seen from the analytical formula that if the value of x is larger and larger, the value of y is smaller and smaller, approaching zero; If x takes a negative value and becomes smaller and smaller, the value of y approaches zero. Therefore, it presents a hyperbola. Similarly, the attributes of images are abstract.
The discussion of image attributes of functions is similar to that of subfunctions.
4. Summary:
Through the study of this lesson, we can understand the concept of inverse proportional function and the properties of its images. We have fully discussed the concept of function and the properties of its image. Mathematics learning requires us to deeply understand, find out the general relationship and development law between things, find problems by mathematical methods, and give some explanations with existing mathematical knowledge. That is, mathematics is a part of the world, but it is also hidden in the world.