Sequence is one of the important contents of high school mathematics. Geometric progression is a new special series after learning from arithmetic progression, which is widely used in life, such as saving and installment payment. In the whole high school mathematics content, series is closely related to the learned function and the limit of series behind it. It is also a good subject to cultivate students' mathematical ability, which can cultivate students' ability to observe, analyze, summarize, guess and solve problems comprehensively. The following are the lecture notes of high school series, welcome to read.

First of all, talk about textbooks.

1, the position and function of teaching materials:

Sequence is one of the important contents of high school mathematics, which not only has a wide range of practical applications, but also has the function of connecting the past with the future. On the one hand, sequence, as a special function, is inseparable from function thought; On the other hand, learning sequence is also a preparation for further learning the limit of sequence. On the other hand, on the basis of students' learning the concept of sequence, arithmetic progression gave two methods of sequence-general formula and recursive formula, which further deepened and broadened his understanding of sequence. At the same time, arithmetic progression also provided a foundation for studying geometric series in the future.

2. Teaching objectives

According to the requirements of the syllabus and the actual level of students, the teaching objectives of this course are determined.

Knowledge: understand and master arithmetic progression's concept; Understand the derivation process and thought of arithmetic progression's general term formula; This paper introduces the idea and method of "mathematical modeling", which can be used.

B. ability: cultivate students' ability of observation, analysis, induction and reasoning; On the premise of understanding the relationship between function and sequence, the method of learning function is transferred to learning sequence to cultivate students' ability of knowledge and method transfer; Improve students' ability to analyze and solve problems through step-by-step exercises.

C. Emotionally: through arithmetic progression's study, cultivate students' spirit of seeking knowledge by taking the initiative to explore and dare to discover; Develop good thinking habits of careful observation, careful analysis and good at summing up.

3. Teaching emphases and difficulties

According to the requirements of the syllabus, I have determined that the teaching focus of this course is:

(1) the concept of arithmetic progression.

② The derivation process and application of arithmetic progression's general term formula.

Because students are unfamiliar with incomplete induction for the first time, it is difficult to deduce arithmetic progression's identity with incomplete induction. At the same time, students are not familiar with the thinking method of "mathematical modeling", and solving practical problems with mathematical thinking is another difficulty of this course.

Second, the teaching method of learning situation:

For the students in Grade Three and Grade One, they have rich knowledge and experience, and their intelligence development has reached the stage of formal operation, and they have strong abstract thinking ability and deductive reasoning ability. Therefore, I pay attention to guidance, inspiration, research and discussion in teaching, so as to conform to the psychological development characteristics of such students and promote the further development of thinking ability.

According to the thinking and psychological characteristics of senior high school students, I use heuristic, discussion and combination of teaching and practice in this class to stimulate students' curiosity by asking questions, so that students can actively participate in mathematics practice activities in the form of independent thinking and mutual communication, and find, analyze and solve problems under the guidance of teachers.

Third, the guidance of speaking and learning:

In the guiding analysis, let students think, let them associate and explore, and encourage students to question boldly, express their opinions around the center, and make clear the thinking methods and problems that need to be solved.

Fourth, talk about teaching procedures.

The teaching process of this lesson consists of (1) review lead-in, (2) new lesson exploration, (3) application examples, (4) feedback exercises, (5) induction, (6) homework and six teaching links.

(1) Review and introduce:

1. From the function point of view, series can be regarded as a series of function values whose definition domain is _ _ _ _ _ _ _ _, so the general term formula of series is _ _ _ _ _ _ _ _. (n﹡； Analytical formula)

Review the last lesson by practicing 1 and prepare for this lesson to study the problem of sequence with function thought.

Xiao Ming knows 100 words at present, and he (she) is going to stop memorizing words from today. As a result, he or she unconsciously forgets two words every day, so in the next five days, his vocabulary will gradually decrease to: 100, 98, 96, 94, 92.

Xiao Fang only knows five words, and he decides to recite 65,438+00 words every day from today, so in the next five days, his vocabulary will increase to 5,654,38+00,654,38+05,20,25.

Through Exercise 2 and Exercise 3, introduce two concrete arithmetic progression, get a preliminary understanding of the characteristics of arithmetic progression, lay the foundation for the later concept learning, create problem situations for learning new knowledge and stimulate students' curiosity. By observing the characteristics of two series, students can draw out the concept of arithmetic progression, and the summary of problems can cultivate students' cognitive ability from concrete to abstract and from special to general.

(B) the new curriculum exploration

1, introducing the concept of arithmetic progression given by nature:

If a series, starting from the second term, the difference between each term and the previous term is equal to the same constant, this series is called arithmetic progression.

This constant is called arithmetic progression's tolerance and is usually represented by the letter D ... Emphasis:

① "From the second item" meets the requirements;

② Tolerance d must be obtained by subtracting the former from the latter;

③ The difference between each item and its previous item must be the same constant (emphasizing "the same constant");

On the basis of understanding the concept, students will transform arithmetic progression's written language into mathematical language and sum up mathematical expressions:

An+ 1-an=d (n≥ 1) At the same time, in order to understand the concept, I found five series, and the students judged whether it was from arithmetic progression or arithmetic progression.

1.9 ,8,7,6,5,4,……; √ d=- 1

2.0.70,0.7 1,0.72,0.73,0.74……; √ d=0.0 1

3.0,0,0,0,0,0,…….； √ d=0

4. 1,2,3,2,3,4,……; ×

5. 1,0, 1,0, 1,……×

First series of tolerances

It should be emphasized that the tolerance can be positive, negative or 0.

2. The second key part is the general formula of arithmetic progression.

In summing up arithmetic progression's general formula, I adopted the discussion teaching method. Given the first tolerance D of arithmetic progression, students discuss the general formula of a4 in groups. Students guess the general formula of a40 by summing up the general formula of A4, and then sum up the general formula of Ann. The whole process is completed by students. Through mutual discussion, students' sense of cooperation is cultivated and teaching difficulties are solved.

If the first term of an arithmetic progression {an} is a 1 and the tolerance is d, it can be obtained according to its definition:

A2-a 1 =d, that is, A2 = a1+d.

A3–A2 = D means: a3 =a2 +d = a 1 +2d.

A4–A3 = D, that is, a4 =a3 +d = a 1 +3d.

……

Guess: a40 = a 1 +39d, and then sum up the general formula of arithmetic progression:

an=a 1+(n- 1)d

At this time, it is pointed out that this method of finding the general term formula is called incomplete induction, and this method of deriving the formula is not rigorous enough. In order to cultivate students' rigorous learning attitude, here is another method to find the general term formula of series-superposition method:

a2–a 1 = d

a3–a2 = d

a4–a3 = d

……

an–an- 1 = d

Adding the left and right sides of the (n- 1) equation respectively, we can get an–a1= (n-1) d, that is, an = a1+(n-1) d (/kloc-).

When n= 1, (1) also holds.

So for all n ∈ n, the above formula holds.

So it is arithmetic progression's general formula {{an}}.

In the process of proving the superposition method, I adopted the heuristic teaching method.

Inspire students to write n- 1 equation with arithmetic progression concept.

Compare the summarized general formula and inspire students to think of adding n- 1 equation. Prove the general formula.

Through this knowledge point, the mathematical ideas of superposition and addition are introduced, and the teaching requirements of "paying attention to methods and highlighting ideas" are gradually achieved.

Then, for example, the first term of a arithmetic progression {an} is 1 with a tolerance of 2, and the general term formula of this series is: an= 1+(n- 1)×2.

That is, an=2n- 1 to consolidate that application of arithmetic progression's general term formula.

At the same time, it is required to draw the image of this series, which shows that arithmetic progression is a linear function of positive integer n, and its image is an infinite number of isolated points evenly arranged. Using the idea of function to study the sequence makes the essence of the sequence clearer.

(3) Application examples

This link is to enable students to enhance their understanding and application of the general formula and improve their ability to solve practical problems through examples and exercises. Through the examples of 1 and 2, it is shown to students that in arithmetic progression's general formula, the relationship among four quantities, a 1, d, n and an, should be viewed from the viewpoint of movement change. When a part of the quantity is known, the other part can be calculated according to this formula.

Example 1 (1) Find the 20th item of arithmetic progression 8, 5, 2, …; Item 30; Item 40

(2) Is-401an item in arithmetic progression? -5, -9,-13, … If yes, which item is it?

The first question, I added 30 or 40 items to strengthen and consolidate arithmetic progression's general formula; The second problem is actually the problem of finding a positive integer solution, and the key is to find the general term formula an of the sequence.

Example 2 in arithmetic progression {an}, it is known that a5= 10 and a 12 =3 1. Find the first term a 1 and the tolerance d.

On the basis of the previous example 1, take example 2 as an exercise to consolidate the general formula.

Example 3 is a practical modeling problem.

When building a house, you should design stairs. It is known that the bottom of the second floor of the building is 3 meters from the ground and the third floor is 5.8 meters from the ground. If the stairs are designed as 16 steps with the same height, what is the height of each step?

For this problem, I adopted a combination of heuristic and discussion teaching methods. Inspire students to pay attention to the "contour" of each step, make students think that the height of each step from the ground constitutes a arithmetic progression, and guide students to turn this practical problem into a mathematical model-arithmetic progression: (Students discuss and analyze, perform the board respectively, and the teacher evaluates the problem. The problems may appear as follows: students think that the number of items is 16, so it is necessary to make it clear that a 1 is the height from the ground of the second floor, a2 is the height from the ground of the first step, and 16 is a 17, so the difficulties can be solved by showing the actual stair diagram by courseware.

The purpose of setting this question is: 1. Strengthen students' comprehensive analysis ability of application problems, 2. Lead out arithmetic progression questions through practical mathematics problems and stimulate students' interest; 3. Furthermore, through mathematical examples, the mathematical thinking method of "starting from practical problems, establishing mathematical models through abstract generalization, and finally restoring" mathematical modeling "to explain practical problems" is demonstrated.

Feedback exercise

Exercise questions 1 and 2 after section 1 (students are required to complete within the specified time). Objective: To familiarize students with the general formula and train their basic skills.

2. Example 3 in the book) The highest step of the ladder is 33cm wide, the lowest step is 1 10cm wide, and there are 10 steps in the middle. The width of each step is arithmetic progression. Calculate the width of the middle level.

Objective: To strengthen the training of students' modeling thinking.

3. If several examples {an} are arithmetic progression, if bn = k an, (k is constant), try to prove that the sequence {bn} is arithmetic progression.

This topic is to train students to improve the sequence problem, learn how to prove the sequence problem by definition, and strengthen arithmetic progression's concept.

(5) Summary (Students summarize the harvest of this class)

1. arithmetic progression's concept and mathematical expression.

Emphasis keyword: starting from the second item, the difference between each item and the previous item is equal to the same constant.

2. arithmetic progression's general formula an= a 1+(n- 1) d will know how to find one of the three.

3. Solve practical problems with the idea and method of "mathematical modeling"

(6) Transfer

Required questions: textbook P 1 14 Exercise 3.2 Questions 2 and 6.

Topic selection: It is known that the first term a 1=-24 of arithmetic progression {an} is a positive number starting from the term 10, so find the range of tolerance D.

(Objective: To improve students' thirst for knowledge through hierarchical homework and meet the needs of students at different levels)

Five, the blackboard design

Highlight the key points of this section on the blackboard, such as the words "from the second item" and "constant" in the definition, and mark them with red chalk, leaving room for students to write questions. The whole blackboard writing fully embodies the teaching method of intensive teaching and more practice.

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