Teaching plan design of basic inequality in senior high school mathematics
Textbook analysis
Based on the systematic study of inequality relations and inequality properties, this lesson has mastered the inequality properties. As one of the important basic inequalities, it lays the foundation for the following research. In order to further understand the nature and application of inequality and study the maximum problem, basic inequality is essential at this time. Basic inequality plays a connecting role in the knowledge system and is widely used in life and production practice, so it is also a good material for educating students on emotional values, and we should focus on basic inequality.
In teaching, we should attach importance to the new curriculum concept. Students should not only accept, memorize, imitate and practice mathematics learning activities, but also explore, practice, cooperate and communicate, read, learn and interact with teachers and students independently. Teachers should play the roles of organizers, guides and collaborators, guide students to participate, reveal the essence and experience the process. Learn from this section to realize that mathematics comes from life and improve the fun of learning mathematics.
Analysis of curriculum objectives
According to the requirements of the new curriculum standards for inequality and the actual situation of students, the following objectives are determined:
1, knowledge and ability goal: understand and master basic inequalities, and use basic inequalities to solve some simple maximum problems; Understand the concepts of arithmetic average and geometric average, learn to construct conditions and use basic inequalities; Cultivate students' ability to explore, analyze and solve problems.
2. Process and Method Objectives: According to the process of creating scenarios, asking questions → analysis, induction, proof → geometric interpretation → application (finding the optimal solution and solving practical problems). Start thinking activities such as observation, analysis, induction, summary and abstract generalization to cultivate students' thinking ability and experience the learning method of mathematical concepts. Through the use of multimedia teaching methods, students are guided to actively explore the essence of basic inequalities, experience the method of learning mathematical laws and experience the fun of success.
3. Emotional and attitudinal goals: 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.
Analysis of Teaching Emphasis and Difficulties
Emphasis: Understand the basic inequality with the combination of numbers and shapes, and explore the proof process and application of the basic inequality from different angles.
Difficulties: 1, three restrictive conditions when the basic inequality holds (referring to one positive, two definite and three phases, etc.);
2. Use basic inequalities to solve the maximum and minimum values in practical problems.
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. The multimedia courseware of modern information technology is used as a teaching aid to deepen students' understanding of basic inequalities.
Teaching preparation
Multimedia courseware and blackboard writing
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:
Create scenarios and ask 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 above picture is the logo of the 24th International Congress of Mathematicians held in Beijing. The logo 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.
[Q] Can you find some equal relations or unequal relations in this picture?
The purpose of this background is to abstract inequalities by using the quantitative relationship between related regions in the graph. On this basis, guide students to understand basic inequalities.
Second, abstract induction:
Generally speaking, for any real number A and B, the equal sign holds if and only if A = B..
[Q] Can you prove it?
Students are writing on the blackboard.
Especially, when a>0, b>0, in the inequality, replace A and B respectively, what do you get?
Design Basis: 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.
Answer:.
Inductive summary
If both a and b are positive numbers, 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.
Third, understanding sublimation:
1, written language description:
The arithmetic mean of two positive numbers is not less than their geometric mean.
2. Basic inequalities understood by combining sequence knowledge.
It is known that A and B are positive numbers, A is the arithmetic mean of A and B, and G is the positive arithmetic mean of A and B. Is there a definite relationship between A and G?
The arithmetic mean of two positive numbers is not less than their positive arithmetic mean.
3, symbolic language narrative:
If yes, yes, if and only if a=b,
Q: 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:
Teaching plan design of basic inequality in senior high school mathematics II
I. teaching material analysis
1, the position and function of this textbook
"Basic Inequality" is the key content of Compulsory 5, which is reflected on the cover of textbooks (showing the covers of textbooks and reference books). It is a further study of inequality on the basis of learning "the nature of inequality", "the solution of inequality" and "linear programming", and is widely used in the process of proving inequality and finding the maximum value. Finding the best value is also a hot spot in the college entrance examination. At the same time, some important mathematical ideas, such as the combination of numbers and shapes, transformation, etc., are infiltrated into this knowledge, which is conducive to cultivating students' good thinking quality.
2. Teaching objectives
(1) Knowledge objective: to explore the process of proving basic inequalities; Will use basic inequalities to solve the maximum problem.
(2) Ability goal: to cultivate students' thinking ability such as observation, experiment, induction, judgment and guess. ?
(3) Emotional goal: cultivate students' rigorous and realistic scientific attitude, experience the harmony and unity of numbers and shapes, appreciate the application value of mathematics, and stimulate students' interest in learning and the spirit of being brave in exploration.
3. Teaching emphases and difficulties
According to the curriculum standards, the following teaching priorities and difficulties are formulated.
Emphasis: apply the idea of combining numbers and shapes to understand inequalities and explore basic inequalities from different angles.
Difficulty: Dig the connotation and geometric meaning of basic inequality, and find the maximum value with basic inequality.
Second, teaching guidance
This lesson uses the geometry sketchpad and multimedia to make an intuitive demonstration. Heuristic teaching method is used to create problem situations and stimulate students to try activities. Practical examples in life are used to let students enjoy solving practical problems. Comparative analysis method is mainly used in class. Let students discuss and comment; Organize students to study, think and practice. Through the harmonious dialogue between teachers and students, emotions can be heard, students' potential and creativity can be maximized, and cognitive benefits can be maximized. Let students love learning, enjoy learning, learning, learning.
Third, study the guidance of law.
In order to better implement the spirit of curriculum reform and reasonably carry out quality education for students, students are always the main body and teachers are the leading factor in teaching. So in teaching, I let students observe and analyze from different angles, guide students to solve problems, feel the formation process of knowledge, cultivate students' consciousness and ability of combining numbers with shapes, and let students learn to learn.
Fourthly, teaching design.
◆ Introduce the logo of the 2002 International Congress of Mathematicians.
◆ Prove the basic inequality by analytical method.
◆ Geometric explanation of inequality
◆ Application of Basic Inequalities
1, introduced with the logo of the 2002 International Congress of Mathematicians.
As the picture shows, this is the logo of the 24th International Congress of Mathematicians held in Beijing. The logo is designed according to the string diagram of Zhao Shuang, an ancient mathematician in China. The bright color makes it look like a windmill, which represents the hospitality of the people of China. (showing windmill)
In the square ABCD, AE⊥BE, BF⊥CF, CG⊥DG, DH⊥AH, let AE=a and BE=b, then the area of the square is S=__, Rt△ABE, Rt△BCF, Rt△CDG and Rt△ADH are congruent triangles.
It is easy to get from the figure that s≥s', that is,
Question 1: Are they equal? When is equality?
Question 2: When A and B are arbitrary real numbers, does the above formula still hold? (Students think positively and help students understand through the geometry sketchpad)
Generally speaking, for any real number A and B, we have.
If and only if (emphasize) a=b, the equal sign holds (reasonable reasoning).
Question 3: Can you prove it? (Let students prove independently)
Design intent
(1) Through the introduction of the emblem of the 2002 International Congress of Mathematicians, students can further understand the long history of mathematics in China and feel the connection between mathematics and life.
(2) Using this icon, it is convenient to observe the relationship between regions and intuitively introduce basic inequalities.
(3) Three thinking questions create scenarios for students to deepen and strengthen their understanding step by step.
2. Prove the basic inequality by analytical method.
If a>0, b>0,
Replace a and b with and respectively. free
It can also be written as
Emphasize that the premise of basic inequality is "positive" (deductive reasoning)
Question 4: Can it be directly deduced from the essence of inequality?
Certificate = 1 GB3 ①
As long as the card = 2 GB3 ②
To prove ②, just prove = 3 GB3 ③.
To prove = 3 GB3 ③, just prove = 4 GB3 ④.
Obviously, ④ holds. If and only if a=b, the equal sign in inequality holds.
(Emphasize the condition of basic inequality "equality")
Design intent
(1) It is proved that the process textbook appears in the form of filling in the blanks, and students can complete it independently, which can further cultivate students' self-learning ability and conform to the spirit of curriculum reform;
(2) Proving process proves the correctness of inequality, which can deepen students' understanding of basic inequality;
(3) This proof method is "analysis method", which will be emphasized in the chapter of "Reasoning and Proof" in the elective textbook. It is necessary for students to get a preliminary understanding here.
3. Geometric explanation of inequality
As shown in the figure, AB is the diameter of a circle, C is any point of AB, AC=a, CB=b, and the intersection point C is the chord DE perpendicular to AB, even AD and BD, then CD= and the radius are.
Question 5: Can you use this diagram to get the geometric explanation of the basic inequality? (Students think positively and help students understand through the geometry sketchpad)
Design intent
Geometric intuition can inspire thinking and help to understand. Therefore, it is an important aspect to learn and understand mathematics intuitively with the help of geometry. Only by understanding intuitively can we really understand.
4. Application of basic inequalities
Example 1. certificate
(Students prove themselves)
Design intent
(1) This example is very simple. Most students will re-prove by imitating the analysis ideas in textbooks, and the process of proving inequalities can be practiced through "analysis".
(2) Students can deepen their understanding of basic inequalities. A and b are not just letters, but symbols. They can be a, b, x, y or polynomials.
(3) This example is not a textbook example, and it is simpler than a textbook example. This step by step will help students understand the connotation of inequality.
Example 2: (1) Write 36 as the product of two positive numbers. When two positive numbers take what value, their sum is the smallest?
(2) Write 18 as the sum of two positive numbers. When two positive numbers take what value, their product is the largest?
(Let the students work in groups to complete the inquiry)
Teaching plan design of basic inequality in senior high school mathematics III
Curriculum standard requirements
Knowledge and skills: learn to deduce and master the basic inequality, understand the geometric meaning of this basic inequality, and master the conditions for the equal sign "≥" in the theorem to take the equal sign: if and only if these two numbers are equal;
Process and method: Explore abstract basic inequalities through examples;
Emotional goal: through the study in this section, I realize that mathematics comes from life and improve my interest in learning mathematics; Memorize, understand and apply comprehensive knowledge points 1:
Basic inequality and its derivation
Process ∨ Knowledge point 2:
The application of basic inequality ∨ goal design 1. By exploring the process of proving inequality from different angles, students can understand the conditions for the establishment of basic inequality and its equal sign;
2. Master the basic inequalities for solving the maximum problem and understand the functions of three restrictive conditions (one positive, two definite and three phases, etc.). ) in solving the maximum problem. Teaching situation 1:
The picture shows the emblem of the 24th International Congress of Mathematicians held in Beijing.
The emblem is designed according to the string diagram of Zhao Shuang, an ancient mathematician in China.
The bright color makes it look like a windmill, which represents the hospitality of the people of China.
Question 1: Can you find some equal or unequal relationships in this pattern?
Analysis: The "windmill" in the figure is abstracted into a graph, and there are four congruent right triangles in the square ABCD. Let the two right-angled sides of a right-angled triangle be A and B, then the side length of the square is.
Teachers guide students to find equal or unequal relations from the area relationship.
We consider that the sum of the areas of four right-angled triangles is, and the area of a square is.
As can be seen from the picture, that is.
When the right triangle becomes an isosceles right triangle, that is, a=b, the EFGH of the square shrinks to a point, and then there is it.
New knowledge: if, then
Teaching situation 2:
First, two square pieces of paper are folded into two isosceles right triangles along their diagonal lines.
Then use these two triangles to construct a rectangle.
The two sides are equal to the right-angled sides of two right-angled triangles, and the excess parts are folded.
Suppose the areas of two squares are and () respectively.
Question 2: Can we find an inequality by investigating the areas of two right-angled triangles and rectangles in the left picture?
New knowledge: if, then
Question 3: Can you prove them by algebra?
Proof: because, that is, (when taking the equal sign)
(In this process, all discoverable values can be real numbers. )
Proof: (analysis): Because, proof is needed,
Prove it to me,
That is to say,
So, (take the equal sign as)
Two important inequalities in blackboard writing
If, then (if and only if, the equal sign holds)
If, then (if and only if, the equal sign holds)
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