The topic I want to talk about today is the sum of the first n terms in geometric series. For this topic, I mainly elaborate from the following six aspects.

First, the structure and content analysis of teaching materials:

The first n sum formula of geometric series is the fifth section of chapter 13 of the second semester of senior two mathematics. The teaching object is senior two students and the teaching time is 2 hours. This class is the first. Prior to this, students have learned the definition of series, the general formula of equal ratio series and equal ratio series, which paved the way for the transition to this section, and this section also laid the foundation for learning the sum and limit of series in the future. This lesson is not only the focus of this chapter, but also the focus of the textbook.

From the overall content of high school mathematics, the chapter "Sequence and Mathematical Induction" is one of the important contents of high school mathematics, which plays an important role in the whole field of high school mathematics. First of all, sequence has a wide range of practical applications. For example, product specification design, saving, installment calculation and so on. Secondly, the preface has the function of connecting the past with the future. Sequence is the continuation of function, which is essentially a special function; Learning sequence lays the foundation for further learning the limit of sequence. Thirdly, sequence is also a good subject to cultivate and improve students' thinking ability. When learning series, we should always observe, analyze and guess, and comprehensively use previous knowledge to solve some problems in series, which is conducive to the improvement of students' mathematical ability.

This section focuses on the first n sum formulas of geometric series and their applications.

The difficulty in teaching is the derivation of the first n terms and formulas of geometric series.

Second, the analysis of teaching objectives:

As a math teacher, we should not only teach students math knowledge, but more importantly, teach students math thoughts and math consciousness. According to the analysis of the structure and content of the above textbooks, and taking into account the psychological characteristics of students' existing cognitive structure, I have formulated the following teaching objectives:

1. Knowledge objective: To understand the derivation method of geometric progression's first n sum formulas and master geometric progression's first n sum formulas and their applications.

2. Ability goal: to cultivate students' ability to observe and think about problems, and to flexibly use basic concepts to analyze and solve problems, and to exercise mathematical thinking ability.

3. Emotional goal: cultivate students' enthusiasm for learning mathematics, and cultivate students' strong will and innovative spirit that are not discouraged when encountering difficulties.

Third, the student situation analysis:

Before studying this section, students have learned the concepts of arithmetic and geometric series and the general formula and the formula of the sum of the first n items of arithmetic series. They have certain mathematical thinking methods, can think about the next content, and are emotionally eager to learn new knowledge.

Fourth, the analysis of teaching methods:

Teaching methods: Mathematics is an important subject to cultivate and develop people's thinking, so in teaching, students should not only "know why" but also "know why". In order to embody the principle of students' development, follow students' cognitive rules and embody the principle of step by step and heuristic teaching, I have made such a teaching design: under the guidance of teachers, I create scenarios, inspire students to think through the setting of open questions, and experience the mathematics contained in the formation of mathematical concepts in their thinking.

This class will adopt the teaching mode of "multimedia optimization combination-motivation-discovery". This model can actively integrate all the elements in the teaching process, such as teachers, students, teaching materials and teaching methods, and create the best teaching atmosphere. It mainly includes heuristic explanation, interactive discussion, research and exploration and feedback evaluation.

Learning style: according to the spirit of the second curriculum reform, changing students' learning style is also an important content of this curriculum reform. As one of the core subjects of basic education, changing students' mathematics learning methods and changing students' passive acceptance of learning into active participation in learning will not only improve students' overall mathematics literacy, but also promote the transformation of students' overall learning methods. In the classroom structure, according to the students' cognitive level, I designed (1) creating scenarios (2) observation and induction (3) discussion and research (4) immediate training (5) summary and reflection (6) task continuation, and six levels of learning methods, which are interlocking and in-depth at different levels, and successfully completed the teaching purpose. Independent exploration, observation and discovery, analogy and conjecture, cooperation and exchange.

Teaching means, using multimedia and POWERPOINT software to assist teaching.

Five, teaching program design:

1, create a scene:

Due to the shortage of funds, a company decided to borrow money from the bank. The two parties agreed that within three years, the company would borrow 654.38 million yuan from the bank every month. In order to repay the principal and interest, the company repaid 654.38+00 yuan to the bank in the first month, 20 yuan in the second month and 40 yuan in the third month. That is, the monthly repayment amount is twice that of last month. Excuse me, if you are a company manager or a bank supervisor, will you sign this contract?

This is a hanging example, and the following "if" brings students into the situation created by the example, so that students can directly participate in the "market economy". According to psychology, situations have suggestive functions. Under the influence of suggestion, students consciously or unconsciously participate in the role in the situation, so their learning enthusiasm and thinking activities will be greatly mobilized.

Introducing themes in this way has the following advantages:

(1) Make use of students' curiosity and take a practical problem as the starting point, so as to arouse students' interest and enthusiasm in learning this lesson.

(2) Learning in the actual situation can enable students to assimilate and index the new knowledge they are learning by using the existing knowledge and experience, so that the acquired knowledge is not only easy to maintain, but also easy to migrate to unfamiliar problem situations.

(3) The content of the question is closely related to the theme and focus of the teaching content of this lesson.

(4) It is conducive to the transfer of knowledge, so that students can clearly understand the practical application of knowledge.

Under the guidance of the teacher, students quickly established two mathematical models of geometric series according to their own knowledge and experience. Sequence {an} is a geometric series with 100000 as the first term and 1 as the common ratio, that is, a constant sequence. Sequence {bn} is a geometric series with 10 as the first term and 2 as the common ratio.

When the students are eager to find the sum of these two series, the introduction of the topic will naturally come. Teachers' enlightenment from special to general, from concrete to abstract, formally introduced the theme.

2, teaching new lessons:

There are two main contents in this lesson, the derivation of the first n-term summation formula of geometric series and the first n-term summation formula of geometric series and its application. The derivation of the first n terms and formulas of geometric series is the difficulty of this lesson. The basis is as follows:

(1) From the cognitive field, in the classification of declarative knowledge, procedural knowledge and strategic knowledge, students need the highest level of strategic knowledge to master strategies and methods.

(2) As far as subject knowledge is concerned, deduction belongs to the "bottleneck" in subject logic. Breaking through this bottleneck will solve the following problems.

(3) Psychologically speaking, students are not familiar with the learning content, and their original knowledge is weak and difficult to understand.

What is said here is how to use multimedia to stimulate and inspire students' thinking and break through the difficulties of teaching materials.

There are two kinds of geometric series: common ratio q= 1 and common ratio q 1.

When q= 1, Sn=na 1.

When q 1, sn = a1+a1q+...+a1qn-1=

How is the result of Q 1 Sn derived? This is the difficulty of this lesson.

Students who have previewed the textbook will know the results and the derivation process, but they don't know why. It can be said that it is difficult for most students to derive this formula according to their own knowledge and experience.

At this time, students can think first. If we use the mathematical thinking method of "from special to general" in mathematics, can we get closer to this result?

We can easily draw the following conclusions:

S 1=a 1，

S2 = a 1+a2 = a 1+a 1q = a 1( 1+q)

S3 = a 1+a2+a3 = a 1+a 1q+a 1q 2 = a 1( 1+q+Q2)

……

sn = a 1+a2+……+an = a 1( 1+q+Q2+……+qn- 1)

Many students may think according to this formula.

a 1( 1+q+Q2+……+qn- 1)= a 1( 1+q+Q2+……+qn- 1)( 1-q)/( 1-q)= a 1

At this time, I want to explain to the students that this thinking method of gradual induction from special to general is very good, and it is a method we often use to solve mathematical problems. Then we should point out that at this stage, we can't prove this process, so its giving is not strict. This not only makes students re-recognize the most basic characteristics and strict logic of mathematics. It also laid the groundwork for learning the content of binomial expansion in the future.

At this point, at this stage, it is impossible to prove that induction is only systematic and rigorous in form, and we can only find another way in the process of thinking. Therefore, we should review arithmetic progression's summation formula, and find a way to deduce the sum formula of geometric progression's first n terms with the help of the thinking method of deducing the summation formula of arithmetic sequence!

Ask the students to recall the derivation process of arithmetic progression's first n terms and formulas.

It can be found that at that time, we interchanged the positions of a 1 and an, a2 and an- 1, which were equidistant from head to tail, and obtained the anti-sum form of Sn. Then add the two formulas. In this way, 2Sn is a constant sequence with n terms, and each term is a 1+an. Thus, the formula of Sn is derived.

Arithmetic progression summation method is based on the characteristics of arithmetic progression and students' knowledge structure and cognitive level. Formally, it is reverse addition. In essence, it is to eliminate the differences between the items in the series, construct a new constant series with the same items, and then deduce the formula of Sn according to the sum of the constant series. Its essential feature is that arithmetic progression starts from the second item with every additional item D.

So can geometric series construct a constant series or a partial constant series in a similar way? Let the students try it themselves. What was the result?

At this time, students will naturally think backwards. The result is obviously not feasible.

At this time, the teacher's main task is to make students' thinking diverge quickly-get rid of the fixed mode of reverse addition. Grasp the students' eagerness to solve this problem and inspire them through the media in time. The teacher should tell the students that the idea of constructing a constant series or a part of a constant series is correct. Since the reverse order can't work, is there any other way to construct a constant sequence?

Then, from the definition of geometric series, guide students to understand geometric series in depth. Starting from the second item, each item is q times that of the previous item, that is to say, after each item is multiplied by q, it becomes its latter item. Then, when both sides of the sum of Sn are multiplied by Q at the same time, the first term in the sum of q Sn is the second term of Sn, which causes the dislocation between Sn and q Sn. Can we construct a sum of a constant sequence or a part of a constant sequence from two sums? How does it add up? Obviously not! Can you subtract? Obviously.

After subtracting Sn from q Sn, a constant sequence in which all terms of N- 1 are 0 is obtained. When we find this constant sequence, the difficulty is broken, the derivation of Sn is easy, and the cognitive goal of this lesson is basically achieved.

In order to deepen our understanding, at this time, we should also make an analogy analysis of the derivation process of the sum formula of arithmetic sequence:

The basic idea of summation of two series is to construct a constant series, and the idea of constructing a constant series is also the basic idea of summation of other series. In the process of constructing constant series, geometric series adopts dislocation subtraction, while arithmetic series adopts reverse addition, which is essentially dislocation addition and is a large-scale dislocation addition, while geometric series is only a small-scale dislocation addition with the step size of 1. It shows that in the sum formula of Sn, multiplying both sides by q at the same time is the key to solving the problem-the key to constructing a constant sequence and the key to deriving the sum formula of an equal ratio sequence.

Therefore, the derivation method of the sum formula of these two series is consistent in mathematical thought and method, but there are also differences, that is, the method of dislocation is different. It is precisely because of this difference that teachers have more teaching space. When teachers lead students out of the thinking formula of "reverse addition", students' thinking quality such as profundity and extensiveness of mathematical thinking is improved, and their thinking quality and thinking ability are improved. In this way, the cognitive goals and quality goals of this class have basically been achieved.

After the formula is derived, it is necessary to explain the characteristics of the formula and let the students remember it. At the same time, another expression of the formula and matters needing attention in application should be explained. Help students understand its form and essence, make clear its connotation and extension, and lay the foundation for flexible use of the formula.

Ask the students to calculate the amount of money in the example by themselves with the sum formula. From the calculation results, let students know that the solution of practical problems can not be separated from mathematics, and they must have a keen mathematical mind in the market economy.

3. Give an example.

When we give examples, we are not only how to solve it, but also why. Summarizing the methods and laws of solving problems in time is helpful to develop students' thinking ability. There are two types of examples in this course:

1) the solution of geometric series to the problem of knowing three and finding two.

Example: Find the first term of geometric series as 2, the sum of the first eight terms with the common ratio of 2, and the value of the fifth term.

And example 4 in the book.

2) Practical application questions.

Example: A sugar factory produces 50,000 tons of sugar in 1 year. If the average annual output is increased by 10% compared with the previous year, the total output can reach 300,000 tons (reserved in one place) in a few years from 1 year?

This setting is mainly based on:

(1) Examples have a corresponding matching relationship with the teaching objectives and tasks stipulated in the syllabus and the key points and difficulties of this lesson.

(2) Follow the principle of consolidation and the idea of teaching-feedback-re-teaching system to set such an example.

(3) Application questions are more suitable for testing intellectual skills, which is conducive to the improvement of mathematical ability. At the same time, it can keep students' interest and initiative in the second half of learning.

4. Formation exercises:

After the example is processed, set up a group of formative exercises as a real-time test of this lesson. The exercises are basically summed directly by formulas, and three exercises are designed according to the cognitive laws and psychological characteristics from easy to difficult and from simple to complex, which is conducive to improving students' enthusiasm. When students practice, teachers patrol, observe the learning situation and get feedback in time. Praise and encourage students' unique solutions in exercises, and discriminate and correct occasional mistakes. Through formative exercises, students' adaptability and ability to draw inferences from others are cultivated, and skills are gradually formed.

5. Course summary

This lesson is summarized as follows:

The first n terms and formulas of (1) proportional series

(2) The derivation method of the formula-dislocation subtraction.

(3) The idea of summation-constructing constant series or partial constant series.

Giving full play to students' main role through teacher-student summary is helpful for students to consolidate their knowledge and cultivate their ability of induction and generalization. Further achieve cognitive goals and quality goals.

Finally, the story of the ancient Indian king Siramu and the inventor of chess ended. The inventor asked the king to put 1 grain in 64 squares on his chessboard, with 2 grains in the second square, 4 grains in the third square and 8 grains in the fourth square ... How many grains should be given to the inventor? Let students feel the wonder of mathematics and stimulate their enthusiasm for learning mathematics.

Homework

According to the difference of students' quality, training at different levels can not only help students master the basic knowledge, but also improve the students with spare capacity, so as to achieve the goal of top-notch and "burden reduction"

You can also arrange corresponding research assignments and think about how to deduce the first n terms and formulas of geometric series by other methods to deepen students' understanding of this knowledge point.

Sixth, teaching evaluation and feedback:

According to the psychological characteristics of senior two students, the content of teaching materials, the principle of teaching students in accordance with their aptitude and inspiring teaching ideas, I adopted the strategy of rule learning and problem solving in this class, that is, "case-formula-application", and the case is a shallow requirement, which gives students a general impression. The formula is an intermediate requirement, from the shallow to the deep, and the key and difficult points are concentrated on derivation and explanation, which is convenient for breakthrough. Application is a comprehensive requirement. We should digest and consolidate what we have learned from different angles and situations, and feedback and verify the implementation of the teaching objectives in this section.

Among them, the case is the foundation, so that students can perceive the teaching materials; Formulas are the key to make students understand textbooks; Practice is application, so that students can consolidate their knowledge and draw inferences from others.

In the three-step teaching, inspiring questions are deduced layer by layer, supplemented by group discussion among students, and intuitive and complete teaching AIDS such as blackboard writing and computer courseware are fully used to change the spoon-feeding teaching mode of teachers' speaking and students' listening, which fully embodies the idea that students are the main body and teachers' teaching serves students. And students use "case-formula-application" from simple to in-depth, from perceptual to rational, from intuitive to abstract.