Teaching plan design of quantity product for high school mathematics plane vector I
Teaching design of plane vector product
Case name design of the first lesson in the third lesson of plane cross product. The analysis of teaching material content plane vector product is the content of the sixth section of chapter 5 of the first volume of senior one of People's Education Press. This class is an important tool to solve some geometric and physical problems. To learn this section, we should master the definition, formula and properties of product of quantities, which is a combination point to examine mathematical ability. Vector models can be established to solve the problems of length, angle, verticality and parallelism in functions, triangles, series, inequalities, analytic geometry and solid geometry, which is an important carrier of "designing propositions on the knowledge network" in college entrance examination propositions. Second, the teaching objectives (knowledge, skills, emotional attitudes, values) (1) knowledge and skills objectives
1, know the generation process of the definition of plane vector product, master its definition and understand its geometric meaning;
2. We can explore the important properties of plane vector product by definition;
3. The product of energy can be used to represent the included angle between two vectors, and the product of energy can be used to judge the vertical and * * * line relationship between two plane vectors.
(2) process and method objectives
(1) through the concept of work that students have learned in physics, guide students to explore the definition of quantity product, and explore the nature by definition;
(2) Derive the geometric meaning of the product of quantities from the physical meaning of work;
(3) Emotion, attitude and values
Through autonomous learning in this section, let students try the process of mathematical research, cultivate students' ability to find, put forward and solve mathematical problems, and help cultivate students' innovative consciousness.
Third, the analysis of learner characteristics Students have learned the basic concepts and knowledge about vectors, and at the same time have certain self-learning ability. Most students have considerable interest and enthusiasm in learning mathematics. However, the development of the ability to explore problems and the awareness of cooperation and communication is not balanced enough and needs to be strengthened. Fourth, the choice of teaching strategies and the design of teaching methods: observation, discussion, comparison, induction, inspiration and guidance.
Learning methods: independent inquiry, cooperation and exchange, induction and summary.
Teacher-student interaction: students explore independently and teachers guide them. Five, teaching environment and resource preparation triangle ruler six, teaching process, teaching process, teacher activities, student activities design intent and resource preparation.
Create scenarios and introduce new courses
Question 1 In physics, we have learned the concept of work. Given the magnitude of force and displacement, can the magnitude of work be calculated? Teacher: Ask questions that students have learned, set questions and stimulate students' interest.
Student: W=FS cos Let students review what they have learned in physics, stimulate students' interest and analyze the form of this formula. Question 2: Who is the angle in the above formula and between whom? How is the angle between two vectors defined? Teacher: Ask the angle and lead to the definition of the included angle between two vectors.
It is pointed out that angle is the angle between force and displacement, and the definition of angle between two vectors can be given through the concept of work in physics and the definition of angle in formula.
Interaction between teachers and students to explore new knowledge
1 leads to the definition of the included angle between two vectors.
Definition: Definition of vector included angle: Let two non-zero vectors a=OA and b=OB, and call ∠AOB= included angle between vectors A and B, (00≤θ≤ 1800).
This concept can be expressed directly to students by teachers in a defined way. )
Teacher: Give the included angle of any two vectors made by students, and guide students to summarize the characteristics and special circumstances of the included angle between the two vectors by drawing.
Student: Students draw pictures. The included angle between any two vectors includes vertical, same direction and reverse direction.
Note: (1) When non-zero vectors A and B are in the same direction, θ=00.
(2) When the directions of A and B are opposite, θ= 1800 (*** straight line or parallel line)
(3) Non-zero vectors such as 0 do not talk about angles.
(4) θ = 900 at ⊥ B.
(5) To find the included angle between two vectors, the two vectors must be translated to a common starting point.
Consolidate the practical application of new knowledge
I will do practical problems.
Example 1 triangle ABC, ∠ABC=450, what is the angle between BA and BC? What about the angle between BA and CB? Students: Cooperate and communicate in groups of four.
Teaching plan design of high school mathematics plane vector quantity product II
First, the general idea:
There are two hidden lines in the design of this lesson: one is to do work around objects in physics and introduce the concept and geometric significance of product of quantities; Secondly, around the concept of product of quantity, new knowledge-calculation formulas of vertical judgment, included angle and line segment length are derived through deformation and limit. Teaching plans can be designed from three aspects: first, the concept of quantity product; The second is the geometric meaning and operation law; The third is the calculation of the modulus and included angle of two vectors.
Second, the teaching objectives:
1. Understand the abstract root of the quantitative product of vectors.
2. Understand the concept of the product of the plane and the included angle of the vector.
3. The relationship between scalar product and vector projection and the geometric significance of scalar product.
4. Understand the properties and operation rules of vector product, and be able to make relevant judgments and calculations.
Third, the key points and difficulties:
Focus 1. Concept and properties of plane vector product.
2. Research and application of arithmetic rule of plane vector product.
Application of vector product of difficult plane
Class arrangement:
2 class hours
Verb (abbreviation of verb) teaching plan and its design intention;
The Physical Background of 1. Plane Vector Product
The quantitative product of plane vector comes from the abstraction of physical problems such as work done by a stressed object in its motion direction. Firstly, it is explained that the displacement of an object placed on a horizontal plane under the action of a horizontal force F is S. There are two vectors in this problem, which is the so-called vector in mathematics. At this time, the work done by the force F of the object is W, where (is the included angle between the vectors F and S, which is the definition basis of the included angle between the two vectors. When defining the angle between two vectors, students should make clear the important condition that "the starting point of vectors is at the same point" and understand vectors. This gives us an inspiration: is work the result of some operation of two vectors? On this basis, the concept of the product of two nonzero vectors A and B is introduced.
Definition of scalar product (inner product) of plane vector
Given two non-zero vectors a and b, the angle between them is θ, then the quantity | a ||| b | cos (called the product of a and b, denoted as a(b), that is, a (b = | a ||| b | cos (,(0≤θ≤π))).
And specifies that the product of 0 and the number of arbitrary vectors is 0.
The direction of zero vector is arbitrary, and its angle with any vector is uncertain. According to the definition of product of quantity, a(b = |a||b|cos) cannot be obtained.
3. The concept of included angle between two nonzero vectors.
Given that non-zero vectors A and B are =a and =b, then ∠AOB=θ(0≤θ≤π) is called the included angle between A and B. 。
, a symbol, is the essence of definition-it is a real number. According to reasoning, when the product of quantity is positive; When the product of quantity is zero; When the quantity product is negative.
4. The concept of "projection"
Definition: |b|cos (called the projection of vector b in direction a).
Projection is also a quantity, and its sign depends on the size of the angle. When (is acute, the projection is positive; (When it is obtuse, the projection is negative; (At right angles, the projection is 0; When (= 0 (the projection is |b| when (= 180), the projection is (| b |). Therefore, the projection can be positive, negative or zero.
According to the definition of the product of quantities, the projection of vector B in direction A can also be written as
Note that the projection of vector A in direction B is different from the projection of vector B in direction A, and it should be distinguished by combining graphics.
5. The geometric meaning of vector product:
The product of quantity a(b) is equal to the product of the length of a and the projection of b in a |b|cos (direction).
The geometric meaning of vector quantity product plays a key role in proving the direction of distribution law. Its geometric meaning is essentially to divide the product into two parts:. This concept is also based on the work done by objects. Is the projection of vector b in direction a.
6. The nature of the product of two vectors:
Let a and b be two nonzero vectors, then
( 1)a(b(a(b = 0；
(2) when a and b are in the same direction, a (b = | a |||| b |; When a and b are opposite, A (B = | A || B |. Special a(a = |a|2 or
(3)|a(b| ≤ |a||b|
(4), where the included angle of non-zero vectors a and b.
Example 1. (1) If the directional quantities A and B are satisfied and the included angle between A and B is 0, then the projection of B on A is _ _ _ _ _ _.
(2) If, then the projection of A in the direction of B is _ _ _ _ _.
Example 2. Known, found according to the following conditions.
Teaching plan design of quantity product of plane vector three in senior high school mathematics
Teaching material analysis:
Cross product's concept is introduced into the textbook on the background that an object is forced to do work. Work is a scalar, which is defined by two vectors: force and displacement, and the reaction is a vector product in mathematics.
The product of vector is a new multiplication that has never been encountered in the past study. It is different from and related to the multiplication of numbers. Through "inquiry", the textbook requires students to use the definition of quantity product of vectors to deduce relevant conclusions. These conclusions can be regarded as a direct inference of the definition.
The first example in the textbook is the direct application of the meaning of quantity product.
Analysis of learning situation:
The concept of vector and the linear operation of vector have been studied before. This paper introduces a new vector operation-vector quantity product. In the textbook, the concept of vector product is introduced on the background that an object is forced to do work, which not only establishes the connection between vector product operation and students' existing knowledge, but also makes students see that the quantity product is related to the size and angle of vector module. At the same time, unlike the previous vector operation, its calculation result is not a vector but a quantity.
Three-dimensional target:
Knowledge and skills
1. Students know and understand the meaning and physical meaning of plane vector product through examples such as "work" in physics, and understand the relationship between plane vector product and vector projection.
2. Students explore three important properties of the product of plane vectors, experience mathematical methods such as analogy and induction, comparison and discrimination, and correctly and skillfully apply the definition and properties of the product of plane vectors to operate.
(2) Process and method
1. Students experience the formation process of mathematical definition from example to abstraction and the discovery process of essence, and further understand the essence of mathematics.
Emotions, attitudes and values
1. Through this course, students can learn mathematics research ideas from special to general and from general to special.
2. Cultivate students' practical operation ability of observing, analyzing and solving problems by solving problems; Cultivate students' communication consciousness and cooperation spirit; Cultivate students' ability to describe and express ideas to solve and explore problems.
Four, teaching difficulties:
1, focusing on the discovery and proof of the concept and properties of plane vector product;
2. Difficulties: understanding the product of plane vector and vector projection;
Prepare teaching AIDS: multimedia, triangle
VI. Schedule: 1 class hour
Seven, the teaching process:
(A) the creation of problem situations, leading to new lessons
Question: Please review, what operations of vectors have we learned? What are the results of these operations?
Introduction to the new lesson: In this lesson, we will learn another operation of learning vectors: the physical background and significance of the product of plane vectors.
New lesson:
1, query the concept of 1: product of quantities.
Show the physical background: The video "Laxmark" abstracts the following physical model from the video.
The first background analysis:
Question: What is the real power that makes the car move forward? What's its size?
A: In fact, it is the component of force in the displacement direction, that is, in mathematics, we named it projection.
The concept of "projection": painting
Definition: | |cos (called the projection of a vector in the direction. Projection is a quantity, not a vector;
2. The second background analysis:
Question: Can you express the "formula for calculating work" in written language?
Analysis: expressed in written language: the work done by a force on an object is equal to the product of the magnitude of the force, the magnitude of the displacement and the cosine of the included angle between the force and the displacement; Work is a scalar, which is determined by two vectors: force and displacement. This gives us a revelation. Can we regard "work" as an operation result of these two vectors?
Definition of scalar product (inner product) of plane vectors: When the sum of two non-zero vectors is known and the included angle is θ, the scalar |||| is called the scalar product of sum, that is, there is |||| (0 ≤ θ≤π), and the scalar product with any vector is designated as 0.
Note: the product of two vectors is a real number, not a vector, and the sign is determined by the sign of cos.
3, the geometric meaning of vector quantity product:
The product of quantity equals the product of the length of and the projection of the direction | |cos (.
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