1, through experiments, statistics and other activities, in which students' awareness and ability of cooperation and communication are further cultivated.

2. Let students learn the estimation methods commonly used in real life through experiments and other activities.

3, combined with the specific situation, preliminary feeling the rationality of statistical inference.

4. Further understand the relationship between probability and statistics.

Teaching emphasis: let students understand the fact that the test frequency approaches the theoretical probability through experiments.

Teaching difficulties: correctly understand that the test frequency approaches the theoretical probability on the premise of multiple tests.

Teaching methods: guiding inquiry and cooperative learning

Teaching aid preparation: courseware

Teaching process:

First, the situation introduction: (multimedia broadcast)

Teacher: How many fish are there in the fish tank?

Health: 3 articles.

Teacher: How many fish are there in the fish pond?

Health: Too many to count.

Teacher: Yes, this is a common problem in our daily life. Some objects can be accurately measured by intuitive counting, while some objects cannot be directly measured by counting because of their large number. In this lesson, we will explore a scientific and effective method with the topic "How many fish are there in the pond".

Second, look for new knowledge.

Teacher: First, let's look at a familiar experiment: (Multimedia Play)

Question 1: There are 8 black balls and 32 white balls in a box. What is the probability of meeting any one? If you randomly find 10, can you infer how many black balls there may be in this 10? Why?

Students think and answer: The total number of balls is 40, and the proportion of black balls is 1/5, so we can pick out 10 at will, and infer that there may be two black balls in this 10.

Question 2: If there are 8 black balls and several white balls in the box, and the balls are not allowed to be poured out, can you estimate the number of white balls? Please design a scheme and give it a try.

Work as a team to accomplish the following tasks:

What experimental method is your group going to adopt? Xiao Ming's or Xiao Liang's? )

② Test times and experimental data were recorded in each group.

Can you estimate the number of white balls in the box according to the data you collected?

Open the box and count the number of white balls in the box. Is your estimate consistent with the actual situation?

Students: (Working in groups) Each group chooses a test method to conduct experiments, statistics and calculations.

Teacher: Give each group timely guidance and evaluation.

Teacher: Please report the test results to each group of students.

1 group: using the first method, it is estimated that there are 52 white balls in the box, which is 2 different from the actual situation;

The second group: using the first method, it is estimated that there are 45 white balls in the box, which is 3 different from the actual situation;

The third group: using the second method, it is estimated that there are 60 white balls in the box, which is 7 different from the actual situation;

Teacher: The error between the estimated value and the actual value obtained through the experiment seems to exist objectively, but it can be confirmed that the test frequency is close to the theoretical probability on the premise of repeating the experiment. Can students analyze which factors will lead to mistakes?

1 group: the balls of two colors are not evenly mixed;

The second group: the shape of the ball is irregular and the mixing is uneven, and the black ball may fall to the bottom and cannot be touched;

The third group: I think the number of tests is the main reason that affects the effect;

…

Teacher: The students' analysis is very good, and there are indeed many factors that lead to mistakes. Therefore, these factors should be considered as much as possible before designing the experiment to reduce the error as much as possible.

Teacher: (Method Construction): Combined with the above experiment: Can you explain the mathematical basis for obtaining the number of white balls? (establish mathematical model)

Student: Group discussion, summary and teacher's comments.

Conclusion: Xiao Ming's method: Suppose there are X white balls in the box, and the total number of balls is x+8. Through many experiments, we can estimate the probability that a ball is randomly drawn from the box and is a black ball. Theoretically, this probability should be equal to, from which the number of white balls X can be estimated.

Xiao Liang's method: Assuming that there are X white balls in the box, we can get the "average level" of the ratio of the number of black balls to the total number of balls in the sample through many sampling surveys, and the "average level" should be similar, so that the value of X can be estimated.

Teacher: Can you talk about the advantages of these two methods?

(Student Debate) Group 1: From the above results, we can see that the result obtained by the first method is closer to the real situation than the second one, and the first method is more accurate.

Group 2: Although the result just now is like this, we think the second method is more scientific, with less tests and less time.

Teacher: I know from the students' arguments why each group of students choose different methods. If you want to investigate how many students in junior high schools in the county buy snacks every day, which method should you choose?

Health: (heated discussion) Choose the second one. The student who just said that the first method is good looks very unhappy.

Teacher: Actually, there is no difference between good and bad methods. As for which method to choose, it should be decided by our research object. If the number of subjects is small and the number of experiments may be enough, it is more accurate to choose the first method. If there are many research objects, such as many practical problems, it will be more convenient and feasible to adopt the second method.

Third, activity exploration:

Teacher: There are only a few white balls in the teacher's box. Without balls of other colors, the balls can't be poured out. How to help the teacher estimate the number of white balls? Communicate with your peers.

Student: After discussion in each group, the answer was quickly found: you can put more black balls in the box, or you can take some balls out of the box and dye them black or mark them in the box, and use the above test method to estimate the number of balls.

Teacher: Great. It seems that the students really understand this method!

Fourth, the application of knowledge:

Now can you design your own plan to estimate the number of fish in the pond?

Scheme: (Student representatives speak orally, teacher comments) You can first fish out a number of fish, mark them, then put them back in the pond for a period of time, and then randomly fish out a number of fish, taking the proportion of marked fish in the whole fish pond as the proportion of marked fish, and then estimate the number of fish in the fish pond.

V. Inductive inquiry:

What are your gains and questions about this lesson? What other problems in life can be solved by today's methods? Give examples.

Health 1: Learned to estimate some practical quantitative problems by sampling inspection;

Student 2: I know that we can choose different experimental methods to estimate the true value according to the number of subjects;

S3: I have a question about how to make the estimation more accurate.

Student 4: I think our group should further strengthen the division of labor and cooperation in the future study, and better complete the task.

This method can be used to estimate how many plastic bags a family throws away a year and the sales volume of a certain commodity.

Health 6: You can estimate how many tits there are in a forest and the yield of a certain kind of rice in an experimental field.

…

Teacher: The students in this class are doing well, and the students in each group have a good division of labor and cooperation. I hope to continue to work hard in the future. The problems in this class need to be discussed in future study. Although our classroom is limited, I hope that students can solve practical problems with what we have learned in the limited classroom in the infinite space of life. I wish every student success in their studies!

Six, homework design (hierarchical):

1, (AB) Consolidate this section and finish the exercises after class.

2.(b) Write down the design plan in detail: How many fish are there in the pond. .

3.(c) Summarize the experimental records of the whole class, and explain the relationship between experimental times and errors through specific calculations.

Seven, deepen the development: (for students with spare capacity to choose to do)

1. Put 10 white balls into a bag full of many black balls and pour out five at a time. Write down the number of white balls poured out and put them back. * * * Down 120 times, down 180 white ball. How many black balls are there in the bag?

Biologists often count the number of birds in a certain area. They captured 100 birds in an area, marked them and put them back in the mountains. After a while, they were captured again. As a result, 5 of the 300 birds caught were tagged. Please estimate how many birds are there in the mountains.

Uncle Zhang wants to know the fish in the pond he contracted. For the first time, he randomly fished out 50 fish with a total weight of 98kg. After marking the 50 fish, he put them back in the pond. After they were completely integrated with other fish, he randomly caught 200 fish with a total weight of 402kg, and there were 5 marked fish. Can you help Uncle Zhang estimate the number and total weight of fish in the pond?

4. Bus No.27 * * * opposite to two districts in a city has two exterior colors, red and green, among which 10 is green. Teacher Zhang often goes to work in a certain community by bus. Can he estimate the total number of buses opposite the urban area and the military area command by an appropriate method? Talk about your opinion.

Teaching reflection: This course takes "student experiment-establishing mathematical model-activity exploration-knowledge application" as the main line, which fully embodies the students' dominant position, truly realizes the change of students' learning style, and the role of teachers as the guide, promoter and collaborator of students' learning is also well reflected. In the future, we should pay attention to strengthening the ability training of group leaders, so that they can better play the role of leading geese in group learning, improve the efficiency of group learning and achieve the purpose of improving classroom teaching efficiency.

How many fish are there in the pond?

Question 1: There are 8 black chess and 32 white chess in a box. What are the chances of randomly picking a black chess player? If you randomly find 10, can you infer how many black cards there may be in this 10? Why?

Question 2: There are eight white balls and several black balls in a pocket. Can you estimate the number of black balls if you are not allowed to reverse the ball?

Put 10 white balls into a bag full of many black balls, and pour out five balls at a time. Write down the number of white balls poured out and put them back. * * * Down 120 times, down 180 white ball. How many black balls are there in the bag?

Experimental reflection:

Question 1: What should we pay attention to in the experiment to make the estimation result more accurate?

Question 2: What are the advantages and disadvantages of the above two methods?

Think about it:

If there are only a few white balls in the box, there are no balls of other colors, and the balls are not allowed to be counted, then how do you estimate the number of white balls?

Classroom exercises:

There are 72 red, yellow and blue glass balls in a pocket. After touching the ball many times, Xiao Ming found that the frequency of touching the red ball, the yellow ball and the blue ball was 35%, 25% and 40% respectively. Try to estimate the number of three kinds of glass balls in your pocket as ().

A 35，25， 12，B 15， 18，39。

C 25， 18，29 D 29，25， 18。

Interdisciplinary synthesis:

Biologists often count the number of birds in an area. They captured 100 birds in an area, marked them and put them back in the mountains. After a while, they were captured again. As a result, 5 of the 300 birds caught were tagged. Please estimate how many birds are there in the mountains.

Use it:

Uncle Zhang wants to know about the fish in the pond he contracted. For the first time, he randomly fished out 50 fish with a total weight of 98kg. After marking the 50 fish, he put them back in the pond. After they were completely integrated with other fish, he randomly caught 200 fish with a total weight of 402kg, and there were 5 marked fish. Can you help Uncle Zhang estimate the number and total weight of fish in the pond?

Application questions:

The appearance colors of long-distance high-speed buses between Jinan and Qingdao are yellow and green, of which 10 is green. Mr. Zhang often travels from Jinan to Qingdao on business and takes a long-distance high-speed bus. Can he estimate the total number of buses leaving Jinan and Qingdao by an appropriate method? Talk about your opinion.

Inductive questions:

What are your gains and questions about this lesson? What other problems in life can be solved by today's methods? Give examples.