Teaching objectives

Further familiar with the content of sine and cosine theorem, can skillfully use cosine theorem and sine theorem to solve related problems, such as judging the shape of triangle and proving triangle identity in triangle.

Emphasis and difficulty in teaching

Teaching emphasis: ingenious application of theorem.

Teaching difficulties: apply sine and cosine theorems to transform the corner relationship.

teaching process

First, review preparation:

1. Write sine theorem, cosine theorem, inference and other formulas.

2. Discuss the triangle type solved by each formula.

Second, teach new lessons:

1. On the solution of teaching triangle;

① Example 1: In △ABC, the following conditions are known to solve triangles.

Two groups of exercises → Discussion: Why does the number of solutions change?

② Analyze and solve with the following figure (when A is an acute angle).

Exercise: In △ABC, the following conditions are known to judge the solution of triangle.

2. Flexible application of sine theorem and cosine theorem in teaching;

① Example 2: In △ABC, it is known that sinA∶sinB∶sinC=6∶5∶4, and the cosine of the angle is found.

Analysis: How do known conditions transform? → Introduce the parameter k, set three sides, and use cosine theorem to find the angle.

② Example 3: In ABC, it is known that a=7, b= 10 and c=6, and the type of triangle is judged.

Analysis: What knowledge can be used to distinguish triangles? → Find the cosine of the angle and judge by the symbol.

③ Example 4: Given △ABC, try to judge the shape of △ABC.

Analysis: How to turn an edge in an angular relationship into an angle? → Rethink: How to Turn Keratosis into Edge?

3. Summary: Discussion on triangular solution; Judging the type of triangle; How to make the corner relationship mutual?

2. Examples of mathematics teaching plans in the second volume of senior three.

First, the analysis of teaching content

The definition of conic curve reflects the essential attribute of conic curve, which is highly abstract after countless practices. Correctly using definitions to solve problems can often control complexity with simple ones. So after learning the definitions, standard equations and geometric properties of ellipse, hyperbola and parabola, I 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 be able to define XX flexibly; 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, Understanding of the Definition of Conic Curve

2. Using the definition of conic curve to find the "maximum"

3, "definition method" to find the trajectory equation

Teaching difficulties:

Defining XX skillfully with conic curve

3. Case analysis of mathematics teaching in the second volume of senior three.

I. Teaching process

1. Review

The concept of inverse function, the solution of inverse function, and the relationship between function domain and value domain of reciprocal inverse function.

Find the inverse function of function y=x3.

2. new lessons.

Let the students draw the image of y=x3 with the geometric sketchpad, and the students begin to draw the image of the function one after another. Some students gave a "Yi" because they got the following image (Figure 1):

The teacher chooses among the students who draw the above images.

Students 1 put their own screen content on other students' screens through the teaching system, and students will respond soon.

Health 2: This is the image of the inverse function of y=x3.

Teacher: Yes, but how did you get this image? Please discuss it.

Students discuss, but can't find the reason. )

Teacher: Let the student 1 show us again so that we can help him find the reason.

(sheng 1 repeated his production process again. )

Health 3: The problem is that he chose it in the wrong order.

Teacher: Which order?

Health 3: Before making point B, when choosing xA and xA3 as the coordinates of B, he first chooses xA3 and then xA, and the coordinates of the point made are (xA3, xA), not (xA, xA3).

Teacher: Is that right? Please let us do it again.

(This time, 1 was selected in the order of xA and xA3, and the image of function y=x3 was obtained. )

Teacher: It seems that the problem really lies in this place. Please think again. Why did he just take the wrong order and get the image of the inverse function y=x3?

The students were thinking again, and then some students raised their hands. )

Teacher: Let's invite Student 4 to tell everyone.

Health 4: Because he did, he just changed the abscissa X of point B (x, y) on y=x3 into the ordinate Y, and the inverse function of y=x3 just changed X into Y. ..

Teacher: That's right. We further study the relationship between the image of y=x3 and the image of its inverse function y=. Can students see the relationship between the images of these two functions?

(Most students answered that the image of y= can be obtained from the image of y=x3, so the teacher asked further. )

Teacher: How to get the image with y= from the image with y=x3?

Health 5: You can get an image with y=x3 by exchanging the abscissa and ordinate of points on the image with y= x3.

Teacher: The abscissa and ordinate are interchanged? How to change it?

The students didn't understand the teacher for a while, and the scene suddenly cooled down, so the teacher had to further clarify the problem. )

Teacher: Actually, I want to ask you if there is a symmetrical relationship between the images of these two functions. If so, what kind of symmetrical relationship is it?

(Students begin to observe the images of these two functions again, and some students raise their hands later. )

Health 6: I found that these two images should be symmetrical about a straight line.

Teacher: Can you tell me which straight line is symmetrical?

Health 6: Not yet.

(Next, the teacher guides the students to find the symmetry axis of the two-function image by using the geometric sketchpad, and draw the following picture, as shown in Figure 2:)

After moving point A (point B and point C move with it), students find that the midpoint M of BC is on the same straight line, which is the symmetry axis of the two-function image. After tracing point M, they found that the locus of the midpoint is a straight line Y = X.

The image of 7: y= x3 and the image of its inverse function y= are symmetrical about the straight line y = x.

Teacher: Is this conclusion general? Do the images of other functions and their inverse functions also have this symmetrical relationship? Please try other features.

(Students draw images of other functions and their inverse functions for verification. Finally, we all come to the conclusion that the image of the function and its inverse function is symmetrical about the straight line Y = X ..)

Some students still raised their hands because they drew the following image (Figure 3):

Teachers have found this problem when they patrol the class. After sending this image to the class, almost everyone can see the problem: the function y=x2(x∈R) in the picture has no inverse function, and ② is not a function image.

Finally, teachers and students sum up:

Point (x, y) and point (y, x) are symmetrical about the straight line y=x;

The image of a function and its inverse function is symmetric about the straight line y = x.

Second, reflection and comment.

1. At the beginning of the semester, I taught Geometry Sketchpad 4. 0, in the teaching process of function image drawing, it is found that students do not pay much attention to the order of abscissa and ordinate when making points according to the selected coordinates, and the design of this lesson stems from this. Although the geometric sketchpad 4. 04, I can draw an image directly according to the resolution function, but this can't reveal the essence of image symmetry, so in this class, I deliberately chose the Geometry Sketchpad 4. 0 is used for teaching.

2. Freudenthal, a Dutch mathematics educator, believes that in the process of mathematics learning, people's thinking process can be guided by vivid and intuitive images, but people's thinking often goes astray because of mistakes in graphics or imagination. Therefore, we should not only rely on intuition, but also get rid of intuition under certain conditions to form abstract concepts. It should be noted that too intuitive examples often affect students' correct understanding of more abstract concepts.

As a tool of modern information technology, computer has strong expressive ability in visualization, for example, in the image and graphic transformation of functions, it can get effects that other intuitive tools can't get; If the computer is only used for intuition, but it can't achieve the purpose of better understanding abstract concepts and promoting students' thinking, then in such teaching, the computer is only an ordinary intuitive tool at most.

In the teaching of this course, computers are more used as tools for students to explore and discover. Students not only find the symmetrical relationship between the function and its inverse function image, but also understand the concept of inverse function at a deeper level and have a deeper understanding of the existence and solution of inverse function.

At present, the main form of computer used in middle school mathematics is assistance, and it is more an intuitive tool and sometimes even an electronic blackboard. The future development direction should be: using computers as students' cognitive tools, let students discover and explore through computers, and even use computers to do mathematics, so as to better understand mathematical concepts, promote mathematical thinking and develop mathematical innovation ability in the process.

3. When drawing the symmetrical relationship between two function images, the question type is not designed properly. Originally, students were supposed to answer the symmetrical relationship between the two function images, but they mistakenly thought that they were asking how to get the image of y=x3, thus going astray. Such problems must be avoided in future teaching.

4. Case analysis of mathematics teaching in the second volume of senior three.

I. Guiding ideology and theoretical basis

Mathematics is an important subject to cultivate and develop people's thinking. Therefore, in teaching, students should not only "know what it is", but also "know why it is". Therefore, we should fully reveal the thinking process of acquiring knowledge and methods under the principle of taking students as the main body and teachers as the leading factor. Therefore, in this class, I focused on the constructivist teaching method of "creating problem situations-putting forward mathematical problems-trying to solve problems-verifying solutions", which mainly adopts the teaching method of combining observation, inspiration, analogy, guidance and inquiry. In teaching methods, multimedia-assisted teaching is adopted to visualize abstract problems and make teaching objectives more perfect.

Second, teaching material analysis

The inductive formula of trigonometric function is the content of the third section of the first chapter of compulsory mathematics 4 in the standard textbook of high school mathematics experiment (People's Education Edition A), and its main content is formulas (2) to (6) in the inductive formula of trigonometric function. This is the first lesson, and the teaching content is Formulas (2), (3) and (4). The textbook requires students to pass the definition and inductive formula of trigonometric function from any angle they have mastered. Using the idea of symmetry, we can find the symmetrical relationship between any angle and the terminal edge, their relationship with the coordinates of the intersection point of the unit circle, and then find the relationship between their trigonometric functions, that is, we can find, master and apply the inductive formulas of trigonometric functions (2), (3) and (4). At the same time, the teaching materials are permeated with mathematical thinking methods such as conversion and conversion, which puts forward requirements for cultivating students' good study habits. For this reason, this section occupies a very important position in trigonometric functions.

Thirdly, the analysis of learning situation.

The teaching object of this class is all the students of Grade One (1) in our school. The students in this class belong to the lower-middle level, but they have a good study habit of being good at hands-on. By using the discovered teaching methods, they should be able to easily complete the teaching content of this class.

Fourth, teaching objectives.

(1). Basic knowledge goal: Understand the discovery process of inductive formulas and master inductive formulas of sine, cosine and tangent;

(2) Ability training goal: correctly use inductive formula to find sine, cosine and tangent at any angle, and evaluate and simplify simple trigonometric functions;

(3) Innovative quality goal: through the derivation and application of formulas, improve the ability of triangle constant deformation, infiltrate the mathematical thought of regression and the combination of numbers and shapes, and improve students' ability to analyze and solve problems;

(4) Personality quality goal: Through the study and application of inductive formula, we can feel the common law of the relationship between things, reveal the essential attributes of things, and cultivate students' historical materialism by using mathematical thinking methods such as transformation.

Key points and difficulties in teaching verbs (abbreviation of verb)

1. Teaching focus

Understand and master inductive formula.

2. Teaching difficulties

Correct use of inductive formula, finding trigonometric function value and simplifying formulas of trigonometric functions.

Six, teaching methods and expected effect analysis

As teachers, we should not only teach students mathematical knowledge, but more importantly, teach students mathematical thinking methods. How to achieve this goal requires every teacher to study hard and explore seriously. I make the following analysis from three aspects: teaching methods, learning methods and expected results.

1. Teaching methods

Mathematics teaching is the teaching of mathematical thinking activities, not just the result of mathematical activities. The purpose of mathematics learning is not only to acquire mathematical knowledge, but also to train people's thinking ability and improve their thinking quality.

In the teaching process of this class, I take students as the theme, take discovery as the main line, try my best to infiltrate mathematical thinking methods such as analogy, transformation and combination of numbers and shapes, and adopt teaching modes such as questioning, inspiration and guidance, joint exploration and comprehensive application to give students "time" and "space", from easy to difficult, from special to general, and try my best to create a relaxed learning environment so that students can experience the happiness of learning and success.

Study law

"Modern illiterates are not illiterate people, but people who have not mastered learning methods." Many classroom teaching methods are often based on high starting point, large capacity and fast progress to teach students more knowledge points, but ignore that it takes time for students to digest knowledge, thus depriving students of their interest and enthusiasm for learning. How to make students digest knowledge to a certain extent and improve their learning enthusiasm is a problem that teachers must think about.

In the teaching process of this class, I guide students to think about problems, discuss them together, solve problems simply, reproduce the exploration process and practice consolidation. Let students participate in the whole process of exploration, let them cooperate, communicate and explore together after acquiring new knowledge and methods to solve problems, and change passive learning into active independent learning.

3. Expected effect

This lesson is expected to help students correctly understand the discovery and proof process of inductive formulas, master inductive formulas, and skillfully apply inductive formulas to understand some simple simplification problems.

5. Examples of mathematics teaching plans in the second volume of senior three.

Teaching objectives

Understand the concept of sequence and master the application of sequence.

Emphasis and difficulty in teaching

Understand the concept of sequence and master the application of sequence.

teaching process

Elaborate knowledge points

1, number series: a series of numbers arranged in a certain order (related to the order).

2. General term formula: The functional relationship between the nth term an and n of the sequence is expressed by the formula: an=f(n).

(general formula is not)

3. Representation of series:

(1) enumeration method: such as 1, 3, 5, 7, 9 ...;

(2) Graphical method: it consists of (n, an) points;

(3) Analysis method: expressed by general formula, such as an=2n+ 1.

(4) Recursive method: each term is represented by the relationship between the first n terms and the values of its neighbors, such as A 1 = 1, AN = 1+2AN- 1.

4. Classification of series: there are finite series and infinite series; Increasing series, decreasing series, swinging series, constant series; Bounded series, XX series

5. The properties of the sum of the first n terms of an arbitrary sequence {an}