Basic Inequality Teaching Plan Model Wen Yi
Teaching objectives
1, knowledge and skills target
(1) Master the basic inequality and know its operation structure;
(2) Understand the geometric and algebraic meanings of basic inequalities;
(3) Basic inequalities can be used to find a simple maximum.
2, process and method objectives
(1) experienced the process of abstracting basic inequalities from geometric figures;
(2) Experience the combination of numbers and shapes.
3. Emotion, attitude and values.
(1) Understand the development process of mathematics and learn to observe and analyze things from a mathematical perspective;
(2) Experience multi-angle exploration and problem solving.
Ability training
Cultivate students' rigorous and standardized learning ability, dialectical analysis ability, application ability and problem-solving ability.
Teaching focus
Using the idea of combining numbers and shapes to understand inequalities, this paper explores the process of proving inequalities from different angles.
Teaching difficulties
Conditions for the equality of basic inequalities.
teaching method
Teachers' inspiration and guidance are combined with students' independent exploration.
teaching tool
Courseware-aided teaching and physics demonstration experiment
teaching process
Shape merging format
Teaching process design
Create scenarios and introduce new courses.
The picture shows the emblem of the 24th International Congress of Mathematicians held in Beijing, which is designed according to Zhao Shuangxian's picture. Demonstrate with the chords stacked before Zhao Shuang's class, and compare the areas of four right-angled triangles with those of a big square. What kind of equal and unequal relationship will you get?
Zhao Shuang's Chord Diagram
1. Explore the inequality relationship in the graph.
The windmill in the picture is abstracted as a right congruent right triangle 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. In this way, the sum of the areas of four right-angled triangles is 2ab, and the area of a square is. Since the area of four right-angled triangles is smaller than that of squares, we get an inequality:.
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.
2. Draw a conclusion: Generally speaking, if.
3. Proof of thinking: Can you give proof of it?
Proof: Because
while
So, that is
4. Basic inequality
1) Especially, if a >；; 0, b>0, we use a, b instead, available, usually we write the above formula:
2) Derive the basic inequality from the nature of inequality.
Through analysis, it is proved that:
Required certificate (1)
Just prove (2)
To prove (2), just prove a+b- 0 (3).
To prove (3), just prove (-) (4)
Obviously, (4) is true. If and only if a=b, the equal sign in equation (4) holds.
3) Understand the geometric meaning of basic inequalities.
Basic inequality teaching plan model II
Subject: 3.4.3 Application of Basic Inequalities (II) Subject: Mathematics Teaching Object: Senior Two (290) Students' Class Time: 1 Class Time Provider: Liu He 'an Unit: Yao 'an No.1 Middle School No.1, Teaching Content Analysis The research of this course is to review and apply the knowledge and methods that students have learned before, and then build a more perfect knowledge network. Cultivation and training of mathematical modeling ability.
According to the teaching content of this lesson, observation, reading, induction, logical analysis, thinking, cooperation and communication, and inquiry are applied to the practical application of basic inequalities, and heuristic inquiry teaching is carried out with the help of projector. Second, the teaching goal (1) knowledge goal: to construct basic inequalities to solve the range and maximum problems of functions;
(b) Ability goal: Let students explore how to solve practical problems with basic inequalities.
(3) Emotion, attitude and values:
Through solving specific problems, let students feel and experience a lot of unequal relations in the real world and daily life and problems that need to be considered from a rational perspective, encourage students to make analogy, induction and abstraction from the perspective of mathematics, let students feel mathematics, enter mathematics, and cultivate students' rigorous mathematical study habits and good thinking habits; ? Third, the analysis of learner characteristics In the teaching process of this course, we should still emphasize the realistic background and practical application of inequality, and truly regard inequality as a tool to describe the inequality relationship in the real world. Through the analysis and solution of practical problems, students can experience the extensive practical value of basic inequalities, and at the same time let students feel the application value of mathematics. So as to inspire students to love and learn mathematics, instead of thinking that mathematics is just a boring reasoning subject. In the process of solving practical problems, students are required to look at many problems in real life from a mathematical perspective, and to deal with many mathematical knowledge and methods such as functions, equations and trigonometry. Fourth, the choice and design of teaching strategies 1. Using inquiry method, according to observation, reading, induction and thinking. ?
2. Teachers provide questions and materials, prompt them in time, and give full play to teachers' leading role and students' main role; ?
3. Design typical challenging questions to stimulate students' positive thinking, so as to cultivate their interest in learning mathematics. V. Teaching emphases and difficulties Teaching emphases: 1. Construct the range and maximum value of the basic inequality solution function.
2. Let students explore how to solve practical problems with basic inequalities; ?
Teaching difficulties: 1. Let students explore how to solve practical problems with basic inequalities; ?
2. Test the conditions for the establishment of the equal sign when applying the basic inequality; ?
Six, teaching process, teacher activities, student activities, design intent (1) to introduce new courses.
(B) to promote the new curriculum
It is known that if ab is a constant k, how does the value of a+b change?
If a+b is a constant s, how does the value of ab change?
The teacher uses the projector to give the first set of questions in this lesson.
(1) Find the function y = 2x2+(x >; 0).？
(2) find the function y = x2+(x >；; 0).？
(3) Find the function y = 3x2-2x3 (0
(4) Find the function y = x (1-x2) (0
(5) Let a>0 and b>0, and a2+ = 1, what is the maximum value?
(3) Cooperative Inquiry Let us consider using the relationship between the arithmetic mean and the geometric mean of positive numbers to answer these questions. According to the meaning of the maximum value of the function, we can easily find that if one end of the average inequality is a constant, then when the equal sign can be obtained, this constant is a maximum value at the other end.
(4) Detailed analysis of examples?
For example, a factory wants to build a cuboid water storage tank with a volume of 4 800 m3 and a depth of 3 m. If the cost per square meter of the tank bottom is 150 yuan and the cost per square meter of the tank wall is 120 yuan, how to design the tank to minimize the total cost? What is the lowest total cost?
If and only if a=b, the minimum value of a+b is 2 K?
If and only if a=b, ab has a maximum (or ab has a maximum).
Students finish
Set aside five minutes for students to think and cooperate.
According to the typical situation of students' performance, find five students to perform on the blackboard, and then the teacher will comment on the students' performance on the blackboard?
Students think and answer,
Basic inequality teaching plan model text 3
First, the background analysis of teaching materials
1. The position and function of teaching materials
The content of this section is developed on the basis of systematically reviewing inequality relations and inequality properties and mastering inequality properties. The textbook reviews the basic inequality through Zhao Shuang's chord diagram. On the basis of algebraic proof, it guides students to review the geometric meaning of the basic inequality through "inquiry", and gives its application in solving the maximum value of function and practical problems, which plays a connecting role in the knowledge system. From the application value of knowledge, basic inequality is a model abstracted from a large number of mathematical problems and practical problems, and the mathematical thinking methods included in formula derivation (such as combination of numbers and shapes, abstract induction, deductive reasoning, analytical proof, etc.). ) widely used in the study of various inequalities; From the humanistic value of the content, the exploration, derivation and application of basic inequalities need students to observe, analyze, guess, summarize and generalize, which is helpful to cultivate students' thinking ability and exploration spirit, and is a good carrier to cultivate students' consciousness of combining numbers with shapes and improve their mathematical ability.
This section is a review class, which not only enables students to further understand the concept, but also can master the application of basic inequality in finding the maximum value and realize the guiding role of basic inequality in real life.
2. Analysis of learning situation
Cognitive aspect, students have mastered the basic properties of inequality, and can compare the size of numbers and formulas according to the nature of inequality, and also have some basic knowledge of plane geometry. How to make students know the word "basic" again is the premise of this lesson. This inequality actually reflects the size change caused by two basic operations of real numbers (namely, addition and multiplication), which is not only reflected in its algebraic structure, but also has geometric significance. The resulting problems have played a very good role in cultivating students' algebraic reasoning ability and geometric intuition ability. Therefore, we must start with the algebraic structure and geometric meaning of basic inequality, so that students can deeply understand its essence.
In addition, when solving the maximum value with basic inequalities, students tend to ignore the preconditions for using basic inequalities and the conditions for the establishment of equal signs. Therefore, in the teaching process, students should fully understand the role of three restrictive conditions (one positive, two definite, three phases, etc.) for the establishment of basic inequalities. ) in solving the maximum problem by discriminating errors.
3. Teaching emphases and difficulties:
Teaching emphasis: understand the basic inequality with the idea of combining numbers and shapes, and review and explore the proof process of the basic inequality from different angles; Solving some simple maximum problems with basic inequalities.
Teaching difficulties: reviewing the process of abstract basic inequalities under the geometric background; The condition of equal sign in basic inequality: applying basic inequality to solve practical problems.
Second, the teaching objectives
1. Review important inequalities and basic inequalities with "Zhao Shuangxian Diagram", and then review the geometric meaning of basic inequalities with "Inquiry" in the textbook. By reviewing the basic inequalities, students can further understand and feel the thinking method of unity of form and number;
2. Through the re-exploration of teaching materials, guide students to develop basic inequalities and experience the application of basic inequalities;
3. Through the variant teaching of examples in textbooks, let students understand and feel the problems that should be paid attention to when applying basic inequalities to find the maximum value, and solve the application of basic inequalities in practice;
4. Stimulate students' enthusiasm for learning mathematics by using the scene of computer screen, and further cultivate students' mathematical application ability;
5. Through students' independent construction of knowledge network structure diagram, deepen the understanding of basic inequalities.
Third, teaching countermeasures
This section is a review of basic inequalities. First of all, with the help of string diagram and geometric sketchpad demonstration, students can review the concept formation process of basic inequality, experience a series of thinking activities such as observation, analysis, guess and popularization of basic inequality model, review the algebraic structure characteristics of basic inequality and experience the method of mathematical abstract thinking; Second, by exploring the proof method of basic inequality and appreciating it from different angles, students can express the structural characteristics of basic inequality in written language, symbolic language and graphic language, and summarize the conditions for the establishment and use of equal signs in basic inequality, so as to further understand the thinking method of combining numbers with shapes; Third, guide students to solve common maxima and practical problems with basic inequalities, and further experience the process of mathematical modeling;
Fourth, the teaching process
(1) Review old knowledge and basic inequalities.
Scene introduction:
Projection shows Zhao Shuangxian's picture.
Question 1. Please review "Zhao Shuang's Chord Diagram" and compare the size of the area S of the square ABCD with the sum of the areas S' of four small triangles to see what kind of inequality can be obtained.
Through the observation of "Zhao Shuang's string diagram", students can know numbers from shapes and get algebraic forms of important inequalities from geometric figures;
If and only if, a=b, an equal sign is obtained. )
Question 3. What are the requirements for real numbers A and B when using basic inequalities?
( )
Please open page 98 of the textbook and look at figure 3.4-3 in the inquiry.
Question 5. Let point d move. Please point out the conditions under which the equal sign holds.
Link 1: Geometric Sketchpad-Drawing by Zhao Shuangxian
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