The second volume of senior two mathematics courseware 1 teaching material analysis.
Students have learned the acute trigonometric function, which is characterized by the ratio of the sides of a right triangle. The introduction of acute trigonometric function is directly related to "solving triangle". Triangular function of any angle is a mathematical model to describe the phenomenon of periodic change, and has nothing to do with "solving triangle". Therefore, like learning other basic elementary functions, the key to learning trigonometric functions from any angle is to let students understand the concept, images and properties of trigonometric functions, and describe some simple periodic changes with trigonometric functions to solve simple practical problems.
In this section, the acute trigonometric function is used as an introduction, and the trigonometric function is defined by using the coordinates of points on the unit circle. Because trigonometric functions are closely related to the unit circle, when we discuss the summation of trigonometric functions, we can get inspiration from the properties of the circle to study what problems and how to study these problems. In the study of trigonometric functions, the idea of combining numbers with shapes plays a very important role.
Analysis of learning situation
Students are familiar with acute trigonometric functions. After extending the angle to any angle, it is difficult to introduce the concept of quadrant angle, and it is also difficult to express the acute trigonometric function by the coordinate ratio of the points on the terminal side of the angle. Before defining trigonometric functions at any angle, you should be fully prepared.
Students may feel uncomfortable when they touch the unit circle for the first time. In teaching, let students realize that it is not only simple and convenient, but also embodies the essence to express the acute trigonometric function with the coordinates of points on the unit circle. Let students realize the convenience of combining numbers with shapes.
Teaching objectives
1. With the help of the unit circle, we can understand the definition of trigonometric function at any angle.
2. According to the definition of trigonometric function, we can understand its domain, the symbol of trigonometric function and the first inductive formula.
3. By studying trigonometric function from any angle, we can further understand the idea of function and the idea of combining numbers with shapes.
4. Let students actively participate in the formation of knowledge, experience the process of knowledge "discovery", gain the experience of "discovery" and cultivate the ability of reasonable guessing.
Teaching emphases and difficulties
Teaching focus
Definition of sine, cosine and tangent at any angle; Symbol of trigonometric function value.
Teaching difficulties
Trigonometric functions are characterized by the coordinates of points on the terminal edge of an angle.
teaching process
Design intent
Teacher-student activities
1, can you recall the definition of acute trigonometric function?
From the original cognitive basis, we can understand the definition of trigonometric function at any angle.
The teacher asked questions and the students answered orally. Then the teacher draws a right triangle.
2. Can we use the coordinates of the point on the terminal edge of the angle in the rectangular coordinate system to represent the acute trigonometric function?
Using human dust to guide students to learn acute trigonometric function.
The teacher establishes an appropriate coordinate system on the plane of the right triangle and draws the terminal edge of the angle; Students give the coordinates of the corresponding points, and use the coordinates to represent the acute trigonometric function.
3. Change the position of the point on the edge of the endpoint. Will these three ratios change? Why?
It shows that these three ratios are independent of the position of the point on the edge of the terminal.
Let the students answer first, and then the teacher guides the students to choose a few points, calculate the proportion, get a specific understanding, and prove it with the nature of similar triangles.
4. Can you simplify the expression by taking appropriate points?
It embodies the simple idea and lays the foundation for drawing the unit circle.
Teachers guide students to compare, and students find that the point with a distance of 1 from the origin can simplify the expression.
5. Define the unit circle.
6. Give the definition of trigonometric function with arbitrary angle.
7. Can you explain the correspondence in the definition?
Through the analysis of correspondence, we can deepen our understanding of the definition.
Teachers guide students to analyze what are the independent variables in the definition of three-solution function, what are the characteristics of the corresponding relationship and what are the function values.
8. Example 1 Example 2 Exercise 1, 2
Deepen the understanding of the definition through examples and exercises.
First, define the idea of solving problems through discussion, and then let students complete the example 1 and exercise 1 independently. Through discussion, determine the solution idea of transforming any point on the unit circle into a point, and then complete Example 2 and Exercise 2.
Reflect on the learning process and summarize the thinking methods of discussing problems.
Let the students sum up first. Teachers should pay attention to ideological content when generalizing on the basis of study and summary, such as the idea of reduction embodied in the process of defining trigonometric function with the coordinates of points on the unit circle, and the idea of guiding trigonometric function research with the concept of general function.
Mathematics Courseware for Senior Two, Volume 2, Teaching Objectives
According to the characteristics of students' cognitive structure and the content of teaching materials, according to the requirements of new curriculum standards, the teaching objectives of this lesson are determined as follows:
(1) Knowledge and skill objectives:
1, understand the significance of the basic theorem of calculus;
2. Newton-Leibniz formula can be used to find simple definite integral.
(2) Process and Method Objective: To realize the method of finding definite integral with the basic theorem of calculus through intuitive examples.
(3) Emotion, attitude and values:
1, learn the dialectical relationship of mutual transformation and unity of opposites between things, and improve rational thinking ability;
2. Understand the scientific value and cultural value of calculus.
3. Teaching emphases and difficulties
Emphasis: Make students intuitively understand the meaning of the basic theorem of calculus and correctly use the basic theorem to calculate simple definite integral.
Difficulty: Understand the meaning of the basic theorem of calculus.
Second, the teaching design
Review:
1, definition of definite integral:
Where-integer,-upper integral limit,-lower integral limit,-integrand function,-integral variable and-integral interval.
2. Geometric meaning of definite integral: Generally speaking, the geometric meaning of definite integral is the algebraic sum of areas between axes, graphs of functions and straight lines. The area above the axis has a positive sign, and the area below the axis has a negative sign.
Curve edge graphic area:;
Variable speed movement distance:;
3, the nature of definite integral:
Course summary:
1. This lesson draws the Newton-Leibniz formula under special circumstances by means of the relationship between the speed and distance of a variable-speed moving object and the graph. Then it is extended to general functions, and the basic theorem of calculus is obtained, and a simple method of finding definite integral is obtained. The key to using this method is to find the original function of the integrand function, which requires everyone to be more skilled in the knowledge of derivative. I hope some students who don't understand will come back to review later!
2. The basic theorem of calculus reveals the internal relationship between derivative and definite integral, and also provides an effective method to calculate definite integral. The basic theorem of calculus is the most important theorem in calculus.