one
I. teaching material analysis
The position and function of teaching materials
The basic inequality, also known as mean inequality, is selected from the third section of chapter 3 of Compulsory Mathematics 5, a standard experimental textbook for ordinary high schools published by Beijing Normal University Press. The teaching object is senior two students. This class is the first class, focusing on the proof of basic inequalities and geometric significance. This lesson is based on the systematic study of inequality relations and the mastery of the essence of inequality. As one of the important basic inequalities, it lays a foundation for further understanding the nature and application of inequalities and studying extreme value problems. Therefore, basic inequality plays a connecting role in the knowledge system and is widely used in life and production practice. It is also a good material to educate students on emotional values, so we should focus on learning basic inequalities.
Teaching objectives
According to the requirements of the new curriculum standards for inequality and the actual situation of students, the following objectives are determined:
Knowledge and skill goals: understand and master basic inequalities, understand the concepts of arithmetic average and geometric average, and learn to construct conditions to use basic inequalities;
Process and Method Objectives: By exploring basic inequalities, students can understand the formation process of knowledge and cultivate their ability to analyze and solve problems;
Emotion and attitude goal: Through the setting of problem situations, let students realize that mathematics comes from reality, cultivate students to see the world with mathematical eyes and understand the world with mathematical thinking, thus cultivating students' good qualities of being good at thinking and diligent in doing things.
Emphasis and difficulty in teaching
Key points: Understand and master basic inequalities, and explain the meaning of basic inequalities with the help of geometric figures.
Difficulty: Deriving inequality by using basic inequality.
The key is to grasp the basic inequality.
Second, the analysis of teaching methods
This lesson adopts observation-perception-abstraction-induction-inquiry; The teaching method of combining inspiration, induction, teaching and practice, with students as the main body and basic inequalities as the main line, starts from practical problems and allows students to explore and think freely. Multimedia-assisted teaching intuitively reflects the teaching content, which fully develops students' thinking activities, thus optimizing the teaching process and greatly improving the efficiency of classroom teaching.
Third, study the guidance of law.
The spirit of the new curriculum reform is based on the development of students, giving them the initiative to learn and advocating active and courageous learning methods. Therefore, this course mainly adopts the learning method of independent exploration and cooperative communication, which enables students to build their own knowledge and become the masters of learning by making them think, do and use.
Fourth, the teaching process
The design of teaching process is problem-centered, focusing on exploring ways to solve problems. This arrangement emphasizes the process, which accords with students' cognitive law, and makes the mathematics teaching process become the process of students' re-creation and rediscovery of knowledge, thus cultivating students' innovative consciousness.
The specific process arrangement is as follows:
(A) the teaching design of basic inequalities creates scenarios and asks questions.
Design intention: Mathematics education must be based on students' "mathematical reality", and the realistic situation problem is the platform of mathematics teaching. One of the tasks of mathematics teachers is to help students construct mathematical reality and develop their mathematical reality on this basis. Based on this, set the following conditions:
The picture above is the emblem of the 22nd International Congress of Mathematicians held in Beijing. The emblem is designed according to the string diagram of Zhao Shuang, an ancient mathematician in China. Light and dark colors make it look like a windmill, which represents the hospitality of the people of China.
[Question 1] Please observe the logo diagram. What are the special geometric figures in the diagram? What is the relationship between their equality and inequality in area? (Let the students discuss in groups)
(B) to explore the problem, abstract induction
Teaching design of basic inequality 1 Explore the inequality relationship in graphics.
Angle of shape —— (Show the changes of logo graphics with multimedia, and guide students to find that the sum of the areas of four right-angled triangles is less than or equal to the square area. )
Angle of number
[Question 2] If the two right-angled sides of a right-angled triangle are A and B respectively, how should this inequality be expressed?
Student discussion results:.
[Question 3] You see, this number is really a bit mysterious. We found an inequality from the graph. Are there any restrictions on the values of a and b here? When does the equal sign in inequality hold? (Teachers and students explore together)
Let's look at the changes in graphics, (teacher demonstrates)
Students found that when a=b, all four right-angled triangles become isosceles right-angled triangles, and the sum of their areas is exactly equal to the area of the square, that is, the exploration conclusion: we get the inequality if and only if the equal sign holds.
Design intention: This background intention is to abstract the teaching design of basic inequalities by using the quantitative relationship between related areas in the diagram. On this basis, guide students to understand basic inequalities.
2. Abstract induction:
Generally speaking, for any real number A and B, the equal sign holds if and only if A = B..
[Question 4] Can you prove it?
Students are writing on the blackboard.
[Question 5] In particular, in the inequality at that time, what did you get by replacing A and B with?
Students can draw such a conclusion.
Design intention: Analogy is an important method to learn mathematics. This link not only allows students to understand the source of basic inequality, but also breaks through the key and difficult points, and also feels the function thought, which lays the foundation for future study.
Inductive summary
If both a and b are non-negative, then the equal sign holds if and only if a = b.
We call this inequality a basic inequality. That is, the arithmetic mean of a and b and the geometric mean of a and b.
3. Explore the proof method of basic inequality:
[Question 6] How to prove the basic inequality?
Design intention: it is to guide students from perceptual knowledge to rational proof and realize the sublimation from perceptual knowledge to rational knowledge. The inequality is obtained from the area relationship in geometry. In the following, we use the idea of algebra and the nature of inequality to directly deduce inequality.
Method 1: Compare or prove the differences through the instructional design of basic inequalities.
Method 2: Analysis method
Yao Zheng
As long as card 2
If you want a certificate, you only need Certificate 2.
If you want a certificate, just a certificate.
Obviously, it is established. If and only if a=b, the equal sign in holds.
4. Understanding sublimation
1) Written language description:
The arithmetic mean of two positive numbers is not less than their geometric mean.
2) Symbolic language narrative:
If yes, yes, if and only if a=b,
[Question 7] How to understand "if and only if"? (Students discuss in groups, exchange views, and summarize by teachers and students)
"If and only if a=b, the equal sign holds" means:
When a=b, take the equal sign, that is;
Only when a=b, take the equal sign, that is.
3) Explore the geometric meaning of basic inequalities:
The teaching design of basic inequality, with the help of geometric figures familiar to junior high school students, guides students to explore the geometric interpretation of inequality, and endows the geometric intuition of inequality through the combination of numbers and shapes. Further understand the conditions of equality in inequality.
As shown in the figure: AB is the diameter of the circle, and point C is a point on AB.
CD⊥AB,AC=a,CB=b，
[Question 8] Can you use this graph to get the geometric explanation of the basic inequality?
(Teachers demonstrate and students feel intuitively)
It is easy to prove RtACDRtDCB, then Cd2 = ca CB.
That is, CD =
The radius of this circle is obviously greater than or equal to CD, that is, if and only if point C coincides with the center of the circle, that is, a = b, the equal sign holds.
Therefore, the geometric meaning of the basic inequality can be considered as follows: in the same semicircle, the radius is not less than the half chord (the diameter is the longest chord); Or think that half of the hypotenuse of a right triangle is not less than the height on the hypotenuse.
4) Basic inequalities understood by combining sequence knowledge.
From the formal point of view, the basic inequality has specific geometric significance; From the point of view of numbers, the basic inequality reveals the unequal relationship between sum and product.
[Question 9] Recall what you have learned. Where do the structures of "sum" and "product" appear?
To sum up:
The algebraic explanation of mean inequality is that the median term of two positive numbers is not less than their median term.
Instructional Design of Basic Inequalities (IV) Experiencing New Knowledge, Migration and Application
Example 1: (1) Let all be positive numbers, and prove inequality: the instructional design of basic inequality.
(2) As shown in the figure: AB is the diameter of a circle, and point C is a point above AB, let AC=a and CB=b,
Can you get the geometric explanation of this inequality with this diagram?
Design intention: The above examples are set according to the difficulties and key points in the use conditions of basic inequalities. The purpose is to further understand the conditions of inequality by using students' original knowledge of plane geometry, and the equal sign is true if and only if. Here, students are completely free to explore, teachers guide and teachers and students summarize.
(E) Practice feedback, consolidate and deepen.
One of the formula applications:
1. Try to judge the relationship with 2?
Question: If the condition "x >；; 0 ",is the above conclusion still valid?
2. Try to judge the relationship with 7?
The second application of the formula:
Design intention: novel and interesting, easy to understand, close to life problems, which not only greatly improves students' interest and broadens their horizons, but more importantly, mobilizes students' interest in exploring and learning, guides students to pay more attention to life, and makes students realize that mathematics is in the life around us.
(1) Weigh an object with a balance with two different arm lengths. Some people say that you only need to weigh it once or so, and then add up the two weighed weights and divide them by 2. Do you think this is lighter or heavier than the actual weight?
(2) Two shopping malls, A and B, promote similar products with the same unit price. The promotion method adopted by a shopping mall is to make a Q discount on the original price of P; The promotion method of Mall B is two discounts. Which discount method is more cost-effective for customers? (0
≠q)
(5) reflect, summarize and integrate new knowledge:
What have you gained from learning this lesson? What lessons have you learned? What other questions do you need to ask?
Design intention: cultivate generalization ability through reflection and induction; Help students sum up experiences and lessons, consolidate knowledge and skills and improve their cognitive level. Summarize the mean inequality from many angles, so that students can grasp the key points of this lesson and break through the difficulties.
Teachers improve as follows according to the situation:
Key points of knowledge:
(1) Conditions and Structural Features of Important Inequalities and Basic Inequalities
(2) The significance of basic inequality in geometry, algebra and practical application.
Thinking methods and skills:
(1) combination of numbers and shapes, "whole and part"
(2) the idea of induction and analogy
(3) substitution method, comparison method and analysis method.
(7) Assign tasks and report to the superior.
1. Preview the teaching design of basic inequalities.
2. Written assignment: Teaching design to prove basic inequalities, knowing that A and B are positive numbers.
3. Thinking: Analogy basic inequality, when A, B and C are all positive numbers, what kind of inequality do you guess?
Design intention: Homework is divided into three forms, which embodies the principle of consolidation and development of homework and considers the differences of students. Reading assignments are the basis of subsequent classes, while thinking questions are not uniformly required for students with spare capacity to study after class.
Evaluation and analysis of verbs (abbreviation of verb)
1. In the process of building new knowledge, teachers strive to guide and inspire students to gradually apply what they have learned to analyze and solve problems, thus forming a more systematic and complete knowledge structure. When designing each question, students' specific conditions are fully considered, so that the questions are accurate and convenient for students to think and answer. Constantly thinking and asking questions in the students' recent development area, students' thinking is valuable, and their understanding and mastery of knowledge are improved and deepened through continuous thinking and discussion.
2. This section requires students to have a full understanding of the basic inequalities of logarithm and form, with special emphasis on the unity of number and form. In the process of teaching, numbers are derived from form and reduced to form, which is intended to make students have a deep understanding of basic inequalities in comparison. As an important mathematical thinking method, "the combination of numbers and shapes" cannot be mastered and applied by teachers. Only after students realize its benefits through practice can students try to use it when solving problems. Only through continuous application can students re-understand this way of thinking, so as to achieve the purpose of mastering it.
Sixth, blackboard design
3.3 Basic inequality
I. Important inequalities
Second, basic inequality.
1. Written language description
2. Symbolic language narrative
3. Geometric significance
4. Algebraic interpretation
Third, the application examples
Example 1.
Fourth, practice feedback.
Verb (abbreviation of verb) summary and induction
1. Knowledge points
2. Thinking method
two
Learning objectives:
1, understand the learning contents and methods in Chapter 2, and describe the definition of random variables.
3. Be able to tell the relationship between random variables and functions; 4. Be able to express random test results with random variables.
Key points: The results of random tests can be expressed as random variables.
Difficulties: thoroughly understand the concept of random events and the purpose of introducing random variables;
Link 1: Definition of random variables
1. Through some random phenomena in life, we can generalize the definition of random variables.
2 can describe the definition of random variables
Can tell the difference and connection between random variables and functions.
Read the questions on page 33 of the textbook first, put forward and analyze the understanding, and answer the following questions?
1. What does it mean to know the law of a random phenomenon?
2. What is the difference between the analysis and understanding of the random test results of two random phenomena? What kind of correspondence has been established?
Summary:
3. Random variables
(1) Definition:
This correspondence is called a random variable. In other words, random variables are made up of every possible result of random experiments.
Map to.
(2) Representation: Random variables are usually represented by capital letters.
(3) Differences and connections between random variables and functions
Function random variable
independent variable
dependent variable
Dependent variable range
The same points are all maps. They are all maps.
Application of Link 2 Random Variables
1, can correctly write all possible results of random phenomena, and can describe random events with random variables.
Example 1: It is known that there are two unqualified products in 10. Three of these 10 products are random phenomena, and the number of defective products contained in them is a random variable. (1) Write all possible results of this random phenomenon; (2) Try to describe the above results with random variables.
Variant: It is known that 2 pieces of 10 products are unqualified. It is a random phenomenon to choose three products from this 10 product. If y represents the number of qualified products in the three products taken out, then try to describe the above results with random variables.
Example 2 Throw a uniform coin twice in a row, and use X to indicate the number of times when the two heads are up, then X is a random change.
Quantities, which respectively represent random events represented by the following sets:
( 1){X=0}(2){X= 1}
(3){X0}
Variant: Throw a uniform coin three times in a row, and use X to represent the times of these three heads-up, then X is a random variable. What are the possible values of x? And explain the random test results represented by these values.
Exercise: Write down the possible values of the following random variables and explain the results of the random variables represented by their values.
(1) The number of times you may encounter a red light when you pass five traffic lights when you come home from school;
(2) A bag contains five balls with the same size, numbered 1, 2, 3, 4, 5. Now three balls are randomly taken out from it, and the numbers of the balls taken out are;
Summary (benchmark)