The teaching goal of geometric series in senior high school mathematics teaching plan.
1. Understand the concept of geometric series, master the general formula of geometric series, and use the formula to solve simple problems.
(1) Understand the definition of geometric series correctly, understand the concept of common ratio, make clear that a series is the limit condition of geometric series, judge that a series is geometric series according to the definition, and understand the concept of equal ratio term;
(2) Correctly understand the usage of the representation of geometric series, and flexibly use the general formula to find the first term, common ratio, number of terms and specified terms of geometric series;
(3) Understanding the essence of geometric series by general formula can solve some practical problems.
2. Through the study of geometric series, gradually cultivate students' thinking qualities such as observation, analogy, induction and conjecture.
3. By summarizing the concept of geometric series, we can further cultivate students' rigorous thinking habits and scientific attitude of seeking truth from facts.
Textbook analysis
(1) knowledge structure
Geometric series is another simple and common series, and its research content is comparable to that of arithmetic progression. Firstly, the definition of geometric series is summarized, and the general term formula is derived. Then, the image is studied, and the concept of equal ratio median term is given. Finally, the application of the general formula is given.
(2) Analysis of key points and difficulties
The focus of teaching is the definition of geometric series and the understanding and application of general formula, while the difficulty of teaching lies in the derivation and application of general formula of geometric series.
(1) Like arithmetic progression, geometric progression is a special series. They have many similar properties, but there are also obvious differences. According to the definition and general formula, the characteristics of geometric series can be obtained, which is the focus of teaching.
(2) Although I have been exposed to incomplete induction in arithmetic progression's study, it is still unfamiliar to students; In the process of deduction, students need to have certain ability of observation, analysis and guess; Whether the first term is established needs to be supplemented, so it is difficult to deduce the general term formula.
③ The comprehensive study of arithmetic progression and geometric progression can not be separated from the general formula, so the flexible use of the general formula is both important and difficult.
Teaching suggestion
(1) It is suggested that this course be divided into two classes, one is about the concept of geometric series, and the other is about the application of geometric series.
(2) With the introduction of the concept of geometric series, we can cite several concrete examples, and students can sum up the same characteristics of these series, so as to get the definition of geometric series. We can also give several arithmetic progression and geometric progression together. Students can classify these series, one of which is divided according to arithmetic and proportional, so that Tsuihiji can sum up the definition of geometric progression.
(3) According to the definition, let students analyze the characteristics that the common ratio of geometric series is not 0 and each term is not 0, so as to deepen their understanding of the concept.
(4) Compared with arithmetic progression's representation, students can summarize geometric progression's representation, inspire students to understand the general formula from the perspective of function, and draw the image of series from the structural characteristics of the general formula.
(5) Thanks to arithmetic progression's research experience, geometric progression's research can be solved by students themselves, and teachers only need to grasp the rhythm of the class and appear as the organizer of a class.
(6) Students can ask questions, solve problems and give lectures to each other, and give full play to students' main role.
Example of instructional design
Title: The concept of geometric series.
Teaching objectives
1. Through teaching, students can understand the concept of geometric series and deduce and master general formulas.
2. Make students further understand the ideas of analogy and induction, and cultivate the ability of observation and generalization.
3. Cultivate students' diligent thinking, seeking truth from facts and rigorous scientific attitude.
Teaching emphases and difficulties
The emphasis and difficulty are the induction of the definition of geometric series and the derivation of general formula.
training/teaching aid
Projector, multimedia software, computer.
teaching method
Discussion and conversation methods.
teaching process
First, ask questions.
Give the following series, classify them and say the classification criteria. (slide)
①-2, 1,4,7, 10, 13, 16, 19,…
②8, 16,32,64, 128,256,…
③ 1, 1, 1, 1, 1, 1, 1,…
④243,8 1,27,9,3, 1,,,…
⑤3 1,29,27,25,23,2 1, 19,…
⑥ 1,- 1, 1,- 1, 1,- 1, 1,- 1,…
⑦ 1,- 10, 100,- 1000, 10000,- 100000,…
⑧0,0,0,0,0,0,0,…
Students express their opinions (according to the relationship between items, they may be divided into increasing series, decreasing series, constant series and swinging series, and may also be divided into arithmetic progression series and proportional series), and a unified classification method is adopted, in which 2346 ⑥ is a series with the same nature (fortunately, students can't see ③, so we will check whether ③ is a geometric series after we get the definition).
Second, explain the new lesson.
Ask the students to say the same characteristics of the sequence 2346. The teacher pointed out that there are many similar examples in real life, such as the problem of amoeba division. Suppose that every unit time passes, each amoeba splits into two amoebas, and then suppose that there is an amoeba at first, and after a unit time, it splits into two amoebas, and after two unit times, there are four amoebas ... Go on, record the number of amoebas in a unit time.
This series also has the same characteristics as the last series, that is, another series we want to study-geometric series (here is the first step to play the multimedia software of the Transformers)
Geometric series (blackboard writing)
1. Definition of Geometric Series (blackboard writing)
According to the difference and connection between geometric series and arithmetic progression's name, this paper tries to define geometric series. Students' general answers may not be perfect. In most cases, students can sum up the foundation of arithmetic progression. The teacher writes the definition of geometric series and marks the key words.
Ask the students to point out the common ratio of 2346⑥ in geometric progression, and think that there are countless columns that are both arithmetic progression and geometric progression. Students can find that ③ is such a series through observation. The teacher will ask if there are any other examples. Let the students give two more examples. Then, ask the students to summarize the general form of this series. Students may say that all shapes of series satisfy both arithmetic and geometric series. Ask the students to discuss and draw a conclusion. At that time, the series was both arithmetic progression and geometric progression, and at that time it was only arithmetic progression.
2. Understand the definition (blackboard writing)
The first term of (1) geometric series is not 0;
(2) Every term of the geometric series is not 0, that is,
Question: If all the items in a series are not zero, what are the conditions for the series to be a geometric series?
(3) The common ratio is not 0.
The definition of geometric series is expressed by mathematical formula.
It is a geometric series (1). The writing of this formula may be controversial, such as the writing.
Let the students study it, ok; Then ask, can you rewrite it into geometric series? Why not? The formula gives the quantitative relationship between the first term and the first term of a series, but can we determine a geometric series? How many conditions do you need to determine a geometric series? Given the first term and the common ratio, how to find the value of either term? So we should study the general formula.
3. General formula of geometric series (blackboard writing)
Question: Use and to represent items.
① Incomplete induction
② Iterative multiplication
,,,, this formula is multiplied, so the general term formula of (blackboard writing) (1) geometric series.
After getting the general formula, let the students think about how to understand the general formula.
(blackboard writing) (2) understanding of the formula
For students, it ultimately boils down to:
① Functional view;
(2) the equation thought (because we already know it in arithmetic progression, review and consolidate it here).
The idea of equation is emphasized here to solve the problem. There are four quantities in the equation, which is the simplest application of the formula. Please give the students an example (you should be able to make up four kinds of questions). What is the format of solving the problem? (Not only to solve problems, but also to pay attention to the training of standardized expression)
If you add a condition, you will know one more quantity, which is a higher level application of the formula. We will learn it in the next class. Students can try to make up some questions.
Three. abstract
1. This lesson learned the concept of geometric series and got the general formula;
2. Pay attention to the analogy with arithmetic progression in research contents and methods;
3. Understand the general formula with the idea of equation and apply it.
Investigation activities
Fold a large piece of tissue paper in half. How thick is it after 30% discount (if possible)? Suppose the thickness of this paper is 0.01mm.
Reference answer:
After 30 times, the thickness is, which exceeds the height of Mount Everest, the highest peak in the world. If the paper is thinner, for example, the paper thickness is 0.00 1 mm, it will exceed the height of Mount Everest if it is folded in half for 34 times. Remember the king's promise? The rice in 3 1 grid is already 107374 1824, and the rice in the following grid is even more. The rice in the last grid should be 6 grains. Use a calculator to calculate (logarithmic calculation is also acceptable).