First, the teaching objectives
(1) Knowledge objective: According to the image of the proportional function, we can observe and summarize the properties of the function; And will be easy to use.
(2) Ability goal: gradually cultivate students' observation ability and generalization ability, discover knowledge through the guidance of teachers, and initially cultivate students' thinking of combining numbers with shapes and mathematical thinking from general to special;
(3) Emotional goal: to stimulate students' interest and enthusiasm in learning mathematics, and gradually cultivate students' scientific attitude of seeking truth from facts.
Second, the focus and difficulty of teaching
Teaching emphasis: the nature and application of proportional function.
Teaching difficulty: discovering the essence of proportional function
Third, teaching methods and learning methods guide teaching methods: guided discovery method and intuitive demonstration method. The difficulty of this lesson is to discover the essence of proportional function. Through the guidance of teachers, students' enthusiasm can be stimulated and mobilized, so that students can do more activities (painting) and observe more images in class, actively participate in the whole teaching activities, and finally discover its essence.
Learning guidance: a learning method that guides students to learn to observe and summarize.
Four, teaching AIDS to prepare computer PPT, onion college computer version five, teaching process:
(1) Review old knowledge and introduce topics.
Review: What is the image of proportional function? ?
A: The image of the proportional function is a straight line passing through the origin (0,0) and the point (1, k).
(2) Know the new:
In two rectangular coordinate systems, draw the image of each group of functions as follows: y=x y=3x? y=4x y=？ y=x？ ②? y=-x？ y=-3x y=-4x？ y=- y=-x
Guide the students to observe the image and see the characteristics of each group of linear distribution. First, draw the image of the above function on the coordinate paper. Then, the onion research institute is used to play the role of the proportional function, and the function image is drawn through dynamic demonstration, thus arousing students' interest in learning and letting them find out the difference between the image they draw and the image in the video.
Observe the image and think about the problem:
1. What is the relationship between the quadrant the image passes through and the value of k? Not clear enough. What is the relationship between the quadrant that the image passes through and the value of k (especially the symbol)?
2. For one of the proportional function images (for example, y=3x), how does the function value y change when x increases? What about x MINUS? Is it necessary to propose a reduction? Please consider.
3. What rules are drawn from it?
The first question: What is the relationship between the quadrant the image passes through and the value of k?
Estimator: It is found that all five straight lines in the first group pass through the first quadrant and the third quadrant; While the five straight lines of the second group all pass through the second and fourth quadrants.
Teacher: From the perspective of scale coefficient, what is the relationship between the scale coefficient of a function and its image distribution? Use words consistently before and after.
Estimated students: the first group k>0, while the second group K.
Teacher: Good. Who can contact them?
Estimator: when k > 0, the function image passes through the first and third quadrants; When k < 0, the function image passes through the second and fourth quadrants.
Teacher: So all the images of the proportional function are true: when k > 0, the images of the function pass through the first quadrant and the third quadrant; When k < 0, does the function image pass through the second quadrant and the fourth quadrant? Computer demonstration: when an image with any positive proportional function moves in the first and third quadrants, the value of k in its analytical formula changes more than zero anyway, and conversely, when the image moves in the second and fourth quadrants, the value of k is less than zero. (This demonstration process can be demonstrated on the website of www.desmos.com/calculator,, so that students can observe the influence of the positive and negative of k on the function image more intuitively. )
The teacher will prove this property: (from observation conjecture to logical proof)
Blackboard writing: when k > 0, the function image passes through the first and third quadrants; When k < 0, the function image passes through the second and fourth quadrants.
Prove: When k>0, if x>0, then kx>0, that is, y>0? The point (x, y) is in the first quadrant.
If x < 0, then kx < 0, that is, y < 0 ∴ the point (x, y) is in the third quadrant.
When x=0, then kx=0, that is, y=0? The point (x, y) is the origin.
That is, all points on the function image (except the origin) are in the first and third quadrants, so the image passes through the first and third quadrants. Similarly, when K.
We can see that when k > 0, the trend of function image is very similar to "Ti" in Chinese character strokes, and when k < 0, the trend is "Si". It's more vivid and easier to remember.
PPT shows the properties of proportional function: when k > 0, the function image passes through the first quadrant and the third quadrant; When k < 0, the function image passes through the second and fourth quadrants.
Teacher: Now do a little exercise, and judge the direction of the function image with the positive proportional resolution function (according to the positive and negative of k).
y=-x y=x？ y= x？ y=-x y=(a2+ 1)x？ (where a is a constant)? y=(-a2- 1)x？ (where a is a constant)
Encourage students to answer enthusiastically.
Conversely, from the quadrant where the function image is located, please name a proportional resolution function that meets the conditions. Ok, let's look at the next question. (The computer reproduces the second question: 2. For one of the directly proportional function images, how does the function value y change when X increases? What about x MINUS? ) Play the onion video.
Blackboard writing: when k > 0, the independent variable X increases gradually, and the function value Y also increases gradually; When k < 0, the independent variable x gradually increases, but the function value y decreases. (i.e. the direction of "Si")
Teacher: Exercise: From the analytic function, please tell its variation: y = 3xy =-xy = xy =-? y=(a2+ 1)x？ (where a is a constant) y = (-a2- 1) x? (where a is a constant)
Encourage students to answer enthusiastically.
The third question: What laws do you draw from it?
Summarize (students answer) the properties of the proportional function y=kx(k≠0);
When k > 0, the function image passes through the first and third quadrants; When the independent variable x increases gradually, the function value y also increases gradually; (that is, the direction of "lifting")
When k < 0, the function image passes through the second and fourth quadrants; When the independent variable x increases gradually, the function value y decreases instead. (i.e. the direction of "Si")
In a word, the nature of the image of the proportional function depends on the symbol of K.
Namely:? K > 0？ (1, 3, increased);
K < 0？ (2, 4, decrease)
(3) Application
1, 0, the analytic formula of the proportional function is _ _ _ _ _ _, and its image must pass? ___________ ? . ?
2, y =- successive images? ___________ ? Quadrant.
3. given AB < 0, the function y=? X image through _ _ _ _ _ _? Quadrant.
4. it is known that the proportional function y = (2a+1) X. if the value of y decreases with the increase of x, find the value range of a. ..
5. When m is what value, y=mxm2-3 is a proportional function, and y increases with the increase of x ...?
Thinking about the problem:
① Given the proportional function y=(m+ 1)xm2+ 1, which quadrants does its image pass through?
② Explain the following proportional functions respectively. When m is a value, does y increase with the increase of x or does y decrease with the increase of x?
a、y=(m2+ 1)x
? b、y=m2x？
? c、y=(m+ 1)x
Summarize this lesson and let us know. ...
Summarizing in the form of a table can sort out the knowledge points and form a network, which is conducive to students' memory and internalization, so that students can sort out the context of knowledge (first play the video, and then summarize the key points of this lesson by PPT).
(5) Exercise on page 89 of homework
(6) Reflection after class
1. Success: The focus of this lesson is the nature and application of proportional function. The difficulty lies in finding the essence of proportional function. Through the guidance of teachers and onion videos, students' enthusiasm can be stimulated, and students can analyze and discover the essence of functions independently. The leading role of teachers and the dominant position of students have been unified. Thus, the focus of this lesson has been highlighted and the difficulties have been broken through; Guide students' study and give feedback; Cultivate students' ability to solve problems by combining numbers with shapes; The teaching of this course focuses on making students "explore, discover and summarize laws independently", which makes it easier for students to understand and master new knowledge and mathematical thinking methods.
2. Disadvantages:
(1) When exploring the properties of the proportional function, it was not predicted that the time for students to draw the function image would be too long, which led to the tension in the later teaching process.
(2) In the process of applying new knowledge, the feedback from students' exercises is not fully understood.
(3) In order to stimulate students' interest in autonomous learning, teachers' classroom language should be refined.
3. Improvement measures:
(1) Have full confidence in students' ability to sum up laws. Give affirmation after students sum up the rules, without too much language repetition, and give students enough space to think and answer questions.
(2) After the students have made clear the nature of the proportional function, they can practice the feedback of new knowledge by taking classroom quizzes, so that teachers can more accurately grasp the students' mastery of new knowledge.
(3) In the process of discovering and summarizing the essence, students should be allowed to do it independently, and teachers need not worry about helping to summarize, so as to concentrate students' attention and stimulate their interest in learning.
In actual teaching, in order to reflect the subjectivity of students' learning and the leading role of teachers' teaching, I spent a lot of time on students' hands-on operation and group discussion, but how to better handle the guidance and explanation of students' exploration process needs constant reflection and progress in actual teaching.