First, the analysis of unit teaching content

The textbook in this chapter is arranged after the trigonometric function chapter, in the middle of compulsory four, paving the way for the later derivation and differential angle formula, and providing convenient tools for solving triangle problems and many calculation problems in plane geometry.

Vector has the characteristics of algebra and geometry, and it is a bridge between algebra and geometry. Vector has algebraic characteristics, and operation and its laws are the basic problems in algebraic research. Vector can perform many operations, such as vector addition, subtraction, multiplication and cross multiplication. Vector operation has a series of rich operational properties. Compared with digital operations, vector operations expand the objects and properties of operations. Vectors have geometric characteristics. It can not only describe and depict points, lines and surfaces in geometry and their positional and quantitative relationships, but also represent curves and surfaces in space. It is a basic tool for studying geometric problems. This textbook can summarize the related concepts of vector from the familiar examples of students through observation, analysis and induction. Compared with previous textbooks, it can make students feel more natural and friendly, help to stimulate students' interest in learning, mobilize their enthusiasm for learning, and make them truly realize it.

Vector is an important mathematical model to describe the real world. It provides a basic mathematical model for understanding abstract algebra, linear algebra and functional analysis. It is closely related to physics. Because vector is an important and basic mathematical concept in modern mathematics and an important tool to communicate algebra, geometry and trigonometric functions, it has a very rich practical background and a wide range of practical applications, so it has high educational and teaching value and is of great significance for updating and perfecting the knowledge structure.

Combined with the geometric background of vector-directed line segment, the textbook introduces the representation method of vector, defines the concept of vector length, and defines the concepts of zero vector, unit vector, parallel vector and * * * line vector. It will be very simple, even very clear, to deal with a lot of old knowledge with vector method, which will help students to have a deeper understanding, stronger memory and more comfortable application of this knowledge. In a word, it is helpful for students to establish a good mathematical cognitive structure. Through this part of the study, students can realize that vectors are closely related to real life and have a wide range of applications in solving practical problems.

Second, the analysis of students in the unit

1. Students have been exposed to vectors in physics in junior high school, and have the basic cognitive level and operational ability, and have the basic ability to explore and discover mathematical conclusions in operations.

2. Students have basically mastered the basic knowledge of functions and trigonometric functions, and will use the methods of combination of numbers and shapes, whole substitution, classified discussion and analogy to solve practical problems.

3. Students have the basic courage and wisdom to analyze and solve mathematical problems.

Third, the teaching objectives

1. Knowledge and skills objectives

(1) Understand and master the basic concept of plane vector, understand the actual background of vector, understand the significance of plane vector and vector equality, and understand the geometric representation of vector through force and force analysis examples.

⑵ Through examples, master the operations of vector addition, subtraction, number multiplication and product of two vectors, and understand their geometric significance.

(3) Understand and master the problems of * * * lines and vertical lines of vectors, understand the basic theorem and significance of plane vectors, master the orthogonal decomposition of plane vectors and their coordinate representation, and use coordinates to represent the addition, subtraction, number multiplication and product operations of vectors.

⑷ Understand the meaning and physical meaning of the scalar product of a plane vector through examples such as "work" in physics, understand the relationship between the scalar product of a plane vector and vector projection, master the coordinate representation of the scalar product, express the included angle between two vectors with the scalar product, and judge the vertical problem of the vector with the scalar product.

2. Process and method objectives

(1) Let students experience the thinking process of observation, analysis, induction and abstract generalization through examples, and feel and understand the vector representation of different dimensions.

⑵ Let students understand the physical and geometric meaning of plane vector product and realize that mathematics and physics are closely related.

⑶ Through the process of solving some simple plane geometric and mechanical problems and other practical problems with vector method, I realize that vector is a tool to deal with geometric and physical problems, which improves students' computing ability and ability to solve practical problems.

3. Emotions, attitudes and values

⑴ Establish the concept of plane vector from life examples that students are familiar with to stimulate students' interest in learning. From the introduction of physical knowledge to the formation of mathematical knowledge, let students realize the interrelationship between knowledge and establish a comprehensive and scientific value view.

⑵ Through orthogonal decomposition of learning vectors, students can further understand that general problems often come down to special problems that people are most familiar with.

Through the study of this chapter, students can realize the connection between mathematics and other knowledge and the role of mathematics as a tool to solve problems.

Key points:

1. The concept of plane vector, operation, * * * line problem, the basic theorem of plane vector.

2. The coordinate representation of plane vector, the concept and properties of vector product, and the vertical problem of vector.

3. The role of experience vector in solving plane geometry problems and physical problems.

Difficulties:

1. Understand the definition of free vector, the addition and subtraction of vector and the basic theorem of plane vector.

2. Understand the coordinate representation of plane vector operation, the concept of vector product, and the application of plane vector product.

3. Represent the geometric relationship with vectors.

Fourth, unit teaching activities.

1. When introducing the concepts related to vectors, students are encouraged to list other examples in real life in addition to the examples given in textbooks.

2. While learning vector knowledge, try to contact familiar physical phenomena or other life examples, express and describe them with vectors, so that students can understand the relationship between knowledge and disciplines.

3. Through collaborative discussion, draw pictures while understanding concepts according to actual cases in life; Drawing while calculating; Further cultivate students' habit of combining numbers with shapes, thinking in images and analyzing problems.

4. In the process of studying this chapter, we should pay attention to two aspects of vector operation: geometric meaning and algebraic representation. Because they are relatively isolated in the process of learning new knowledge, it is not easy for students to form a system of understanding them. Therefore, we should consciously infiltrate and pave the way when teaching new classes, and emphasize their differences and connections when summing up chapters, so that students can understand vectors more comprehensively and deeply.