The first part: Teaching plan template for senior high school mathematics preparation 1. Preview target
Preview the application examples of plane vector, realize that vector is a tool to deal with geometric problems and physical problems, and establish the relationship between actual problems and vectors.
Second, preview the content
Read the textbook, arrange examples, and solve practical geometric and physical problems with vector operation. In addition, think about several issues:
1, for example 1 If vector method is not used, is there any other method to prove it?
2. What is the trilogy of solving plane geometry problems by vector method?
3. In Example 3,
(1) Why is |F 1| the minimum value? What's the minimum?
(2) Can | f1| be equal to |G|? Why?
Third, ask questions.
Students, through your independent study, what doubts do you have? Please fill in the form below.
Classroom inquiry learning plan
First, the learning content
1. Use the knowledge of vectors (vector addition and subtraction, vector quantity product, etc.) to solve the problems of parallelism, perpendicularity, equality, included angle and distance of lines or segments in plane geometry and analytic geometry. ).
2. Use vector knowledge to solve simple physical problems.
Second, the learning process
Inquiry 1:
How to treat the analogy between (1) vector operation and the conclusion "If, then, parallel or coincident with a straight line" in geometry?
(2) Several geometric examples are given by linear operation.
Example 1. It is proved that the sum of squares of two diagonals of parallelogram is equal to the sum of squares of four sides.
It is called parallelogram ABCD.
Verification:
Try to solve this problem by geometric method, and solve the trilogy of plane geometry problem by vector method?
(1) Establish the relationship between plane geometry and vector,
(2) Studying the relationship between geometric elements through vector operation,
(3) Transform the operation results into geometric relations.
Example 2 As shown in the figure, in the parallelogram ABCD, point E and point F are the midpoints of AD and DC sides respectively, and BE and BF intersect with AC at point R and point T respectively. Can you find the relationship between AR, RT and TC?
Question2: When two people carry a travelling bag, the greater the included angle, the more laborious it is. Pull-ups on the horizontal bar, the smaller the angle between the two arms, the more labor-saving. What's wrong with these forces?
Example 3, in daily life, have you ever had such an experience: two people carrying a travel bag, the bigger the angle, the more laborious it is; Pull-ups on the horizontal bar, the smaller the angle between the two arms, the more labor-saving. Can you explain this phenomenon mathematically?
Please consider the following questions in combination with the questions just now:
(1) Why is |F 1| the minimum value? What's the minimum?
(2) Can | f1| be equal to |G|? Why?
As shown in the picture, the two banks of a river are parallel, the width of the river is m, and a ship sets out from A to the other side of the river. Given the ship speed |v 1|= 10km/h and the current speed |v2|=2km/h, ask how long it will take for the shortest voyage (accurate to 0. 1 min)?
Variant training: two particles A and B are emitted from the same source. At a certain moment, their displacements are, (1). Write the displacement S of particle B relative to particle A at this moment; (2) Calculate the projection of S in the direction.
Third, reflection and summary.
Combined with the characteristics of graphics, orthogonal bases are selected, and vectors are represented by coordinates to solve geometric problems and reflect geometric problems.
The characteristics of algebra and the mathematical thought of combining numbers and shapes are vividly reflected. As a bridge tool, vector makes the operation concise and beautiful, which embodies the beauty of mathematics. This method is often used for parallel and vertical problems such as rectangles, squares and right triangles.
This section mainly studies how to solve plane geometry problems and physical problems with vector knowledge. Master vector method and coordinate method, and the steps to solve practical problems with vectors.
The second part: the content analysis of high school mathematics lesson preparation template;
1, set is an important basic concept in middle school mathematics.
The initial concept of set is permeated in primary school mathematics, and some problems are further expressed in the language of set in junior high school. For example, there are number sets and solution sets used in algebra; A set of points used in geometry. As for logic, it can be said that learning mathematics from the beginning is inseparable from mastering and applying logic knowledge. Basic knowledge of logic is also an indispensable tool for understanding and studying problems in daily life, study and work. These can help students understand the significance of learning this chapter, which is also the basis of learning this chapter.
The reason why the collection of preparatory knowledge and simple logic knowledge is arranged at the beginning of high school mathematics is because in high school mathematics, these knowledge are closely related to other contents and are the basis for learning, mastering and using mathematical language.
For example, the next chapter talks about the concept and properties of functions, which cannot be separated from sets and logic.
This section starts with the examples of set in algebra and geometry in junior high school, leads to the concepts of set and elements of set, and illustrates the concept of set with examples.
Then it introduces the common representation methods of sets, including enumeration method and description method, and gives a drawing example to represent sets.
This lesson mainly studies the introduction of the whole chapter and the basic concepts of set.
The purpose of learning introduction is to arouse students' interest in learning and let them know the significance of learning this chapter.
The teaching focus of this lesson is the basic concept of set.
Set is an original and undefined concept in set theory.
When I came into contact with the concept of set, I got a preliminary understanding of the concept mainly through examples.
The textbook gives that "generally speaking, some specified objects become a set together, also known as a set."
This sentence is only a descriptive explanation of the concept of set.
Teaching process:
First, review the introduction:
1. Introduce the development of number sets, and review the greatest common divisor and the least common multiple, the sum of prime numbers;
2. Introduction of chapters in the textbook;
3. The founder of set theory-Cantor (German mathematician) (see appendix);
4. "Birds of a feather flock together" and "Birds of a feather flock together";
5. Examples in textbooks (P4).
Second, explain the new lesson:
Read the first part of the textbook and ask the following questions:
What are the concepts of (1)? How is it defined?
(2) What symbols are there? How is it expressed?
(3) What are the characteristics of the elements in the set?
The concept of (1) set: it is composed of some numbers, some points, some graphs, some algebraic expressions, some objects and some people. We say that all the objects in each group form a set, or that some specified objects together form a set, which is also called a set for short. Every object in a set is called an element of this set.
Definition: Generally, some specified objects are brought together to form a set.
1, the concept of set
(1) Set: Set some specified objects together to form a set.
(2) Element: Each object in a set is called an element of this set.
2, commonly used digital sets and symbols
(1) nonnegative integer set (natural number set): the set of all nonnegative integers, denoted as n, N={0, 1, 2, …}
(2) Positive integer set: the set excluding 0 in the non-negative integer set, which is recorded as N* or N+, where n * = {1, 2,3, ...}
(3) Integer set: the set of all integers, denoted as z, z = {0, 1, 2, …}
(4) Rational number set: the set of all rational numbers, denoted as q, Q={ integer and fraction}
(5) Real number set: the set of all real numbers, denoted as r, where R={ the number corresponding to all points on the number axis}
Note: (1) natural number set is the same as the non-negative integer set, that is, natural number set contains the number 0.
(2) A set that does not contain 0 in a non-negative integer set is recorded as N* or N+
Other sets of numbers, such as Q, Z and R, which do not contain 0, are also expressed in this way. For example, a set that does not include 0 in an integer set is denoted as Z*
3. The connection between elements and sets
(1) belongs to: If A is an element of the set A, it is said that A belongs to A and marked as A ∈ A.
(2) Does not belong to: If A is not an element of the set A, it is said that A does not belong to A and is recorded as aA.
4. Characteristics of elements in the set
(1) Certainty: Given an element in this set, it cannot be ambiguous according to clear criteria.
(2) Reciprocity: the elements in the set are not repeated.
(3) Disorder: The elements in the set have no certain order (usually written in normal order).
5.( 1) sets are usually represented by capitalized Latin letters, such as A, B, C, P and Q. ...
Elements are usually represented by lowercase Latin letters, such as A, B, C, P, Q. ...
(2) The opening direction of "∈" cannot be written with a∈A backwards.
The third part: the teaching goal of high school mathematics lesson preparation template;
1. Understand the geometric meaning of complex numbers and express them with points and vectors in the complex plane; Understand the geometric meaning of addition and subtraction in complex algebraic form.
2. By establishing the one-to-one correspondence between points on the complex plane and complex numbers, the geometric meaning of complex number addition and subtraction is explored independently.
Teaching focus:
Geometric meaning of complex numbers, geometric meaning of addition and subtraction of complex numbers.
Teaching difficulties:
Geometric significance of complex addition and subtraction.
Teaching process:
First, the problem situation
We know that there is a one-to-one correspondence between real numbers and points on the number axis, and real numbers can be represented by points on the number axis. So, can complex numbers also be represented by points?
Second, student activities.
Question 1 Any complex number a+bi can be uniquely determined by an ordered real number pair (a, b), and the ordered real number pair (a, b) corresponds to the points in the plane rectangular coordinate system one by one. So how can we use points on the plane to represent complex numbers?
Question 2: Point A in the plane rectangular coordinate system corresponds to the vector from the origin O to the end of A, so can complex numbers be represented by plane vectors?
Question 3: Any real number has an absolute value, which represents the distance from the point corresponding to this real number to the origin on the number axis. Any vector has a modulus which represents the length of the vector. Based on this, can we give a concept of the modulus (absolute value) of a complex number? What is its geometric significance?
Question 4: Complex numbers can be represented by vectors on the complex plane. So, what is the geometric significance of the addition and subtraction of complex numbers? Can it be obtained as a graphic method like vector addition and subtraction? What is the geometric meaning of the difference between two complex numbers?
Third, structural mathematics.
1. Geometric meaning of complex number: In the plane rectangular coordinate system, the real part A of complex number a+bi is taken as the abscissa and the imaginary part B as the ordinate, and the Z(a, b) point is determined. We can use point Z(a, b) to represent complex number a+bi, which is the geometric meaning of complex number.
2. Complex plane: establish a rectangular coordinate system to represent the plane of complex numbers, where the X axis is the real axis and the Y axis is the imaginary axis. All points on the real axis represent real numbers, and all points on the imaginary axis represent pure imaginary numbers except the origin.
3. Because the point Z(a, b) on the complex plane corresponds to the vector with the origin o as the starting point and the end point z as the end point, we can also use the vector to represent the complex number z=a+bi, which is also the geometric meaning of the complex number.
4. The geometric meaning of complex number addition and subtraction can be obtained by the parallelogram rule of vector addition and subtraction, and the differential modulus of two complex numbers is the distance between two points corresponding to these two complex numbers on the complex plane. At the same time, the law of complex addition and subtraction is completely consistent with the coordinate form of plane vector addition and subtraction.
The fourth part: the template of senior high school mathematics lesson preparation 1. Teaching content analysis
The definition of conic curve reflects the essential attribute of conic curve, which is highly abstract after countless practices. When solving problems properly, simplicity can control complexity in many cases. Therefore, after learning the definitions, standard equations and geometric properties of ellipse, hyperbola and parabola, we should emphasize the definition again and learn to skillfully use the definition of conic curve to solve problems. "
Second, the analysis of students' learning situation
Students in our class are very active and active in classroom teaching activities, but their computing ability is poor, their reasoning ability is weak, and their mathematical language expression ability is also slightly insufficient.
Third, the design ideas
Because this part of knowledge is abstract, if we leave perceptual knowledge, it is easy for students to get into trouble and reduce their enthusiasm for learning. In teaching, with the help of multimedia animation, students are guided to find and solve problems actively, actively participate in teaching, find and acquire new knowledge in a relaxed and pleasant environment, and improve teaching efficiency.
Fourth, teaching objectives.
1. Deeply understand and master the definition of conic curve, and can flexibly apply the definition to solve problems; Master the concepts and solutions of focus coordinates, vertex coordinates, focal length, eccentricity, directrix equation, asymptote and focal radius. Can combine the basic knowledge of plane geometry to solve conic equation.
2. Through practice, strengthen the understanding of the definition of conic curve and improve the ability of analyzing and solving problems; Through the continuous extension of questions and careful questioning, guide students to learn the general methods of solving problems.
3. With the help of multimedia-assisted teaching, stimulate the interest in learning mathematics.
Five, the teaching focus and difficulty:
Teaching focus
1. Understand the definition of conic.
2. Using the definition of conic curve to find the "maximum"
3. "Definition method" to find the trajectory equation
Teaching difficulties:
Clever use of conic definition to solve problems
Chapter 5: Teaching plan template for senior high school mathematics preparation 1. Teaching objectives
1. Knowledge and skills
(1) Master the basic skills of drawing three views.
(2) Enrich students' spatial imagination
2. Process and method
Mainly through students' own personal practice and drawing, we can understand the role of the three views.
3. Emotional attitudes and values
(1) Improve students' spatial imagination.
(2) Experience the function of three views.
Second, the focus and difficulty of teaching
Point: Draw a simple assembly of three views.
Difficulties: Identify the space geometry represented by three views.
Third, learning methods and teaching tools.
1. Learning methods: observation, hands-on practice, discussion and analogy.
2. Teaching tools: physical model, triangle.
Fourth, teaching ideas
(A) the creation of scenarios to uncover the theme
"Viewing the peak from the ridge" means that the same object may have different visual effects from different angles. To truly reflect an object, you can look at it from multiple angles. In this lesson, we mainly study three views of space geometry.
In junior high school, we learned three views (front view, side view and top view) of cube, cuboid, cylinder, cone and sphere. Can you draw three views of space geometry?
(b) painting practice.
1. Put the ball and cuboid on the platform and ask the students to draw three views of them. Teachers will patrol, and students can exchange results and discuss after drawing.
2. Teachers guide students to draw three views of a simple assembly by analogy.
(1) Draw three views of the ball on the cuboid.
(2) Draw three views of the mineral water bottle (the object is placed on the desktop)
After painting, students can show their works, communicate with their classmates and sum up their painting experience.
Before making three views, you should carefully observe and understand its basic structural characteristics before drawing.
3. The mutual transformation between three views and geometry.
(1) Display pictures by projection (textbook P 10, figure 1.2-3)
Ask the students to think about the geometry represented by the three views in the picture.
(2) Can you draw three views of the truncated cone?
(3) What is the role of three views in understanding space geometry? What experience do you have?
Teachers patrol the guidance, answer students' learning difficulties, and then let students express their views on the above issues.
Please draw three views of space geometry represented by other objects in 1.2-4 and communicate with other students.
(3) Consolidate exercises
Textbook P 12 exercise 1 2
P 18 exercise 1.2A group 1
(4) inductive arrangement
Ask the students to review and publish how to make three views of space geometry.
Extracurricular exercises
1. Make a triangular pyramid model with quadrangular bottom and congruent sides, and draw its three views.
2. Make a prism model with similar top and bottom surfaces and congruent isosceles trapezoid sides, and draw its three views.