Teaching objectives
1, mathematical knowledge: master the concept, general formula and related properties of geometric series;
2. Mathematical ability: through the analogy study of arithmetic progression and geometric progression, cultivate students' analogy induction ability;
Mathematical research method of inductive conjecture proof;
3. Mathematical thinking: cultivate students' mathematical thinking of classified discussion and function.
Emphasis and difficulty in teaching
Emphasis: the concept of geometric series and its general formula, how to learn geometric series by analogy with arithmetic progression;
Difficulties: the exploration process of the essence of geometric series.
teaching process
Teaching process:
1, problem introduction:
We have studied a special series arithmetic progression.
Question 1: What is the condition of arithmetic progression? How to determine a arithmetic progression?
(Students dictate and project): If a series starts from the second item, and the difference between each item and the previous item is equal to the same constant, then the series is called arithmetic progression.
To determine a arithmetic progression, we only need to know its first term a 1 and tolerance d.
Arithmetic progression's first terms a 1 and D are known, so arithmetic progression's general term formula is: (blackboard writing) an = a1+(n-1) D.
Teacher: Actually, what is the key to arithmetic progression? Poor? In other words, if a series, starting from the second item, the difference between each item and its previous item is equal to the same constant, then this series is called arithmetic progression.
Similarly, we ask such a question.
Question 2: If the sum of each term and its previous term is equal to the same constant from the second term, a series is called a series.
Fill in the blanks here to guide students to play their creativity. And then what? With what? Product? The situation can be illustrated by concrete examples: if a series starts with the second item, what about each item and its previous item? And then what? (or? Product? ) is equal to the same constant, and this series is recursive? Periodic sequence? And what is most similar to arithmetic progression? Than? Are the same constant. And this series is the geometric series we are going to study today. )
2. New lesson:
1) Definition of geometric series: If a series starts from the second term and the ratio of each term to its previous term is equal to the same constant, then this series is called geometric series. This constant is called the common ratio.
Teacher: This involves the general formula of geometric series. How did you get arithmetic progression's general formula? Similar to arithmetic progression, what do you need to know to determine the general formula of a geometric series?
Teachers and students briefly review arithmetic progression's general formula derivation methods: accumulation method and iteration method.
Formula derivation: (both teachers and students * * *)
If the common ratio of geometric progression is q and the first term is a 1, then:
Method 1: (Multiplication)
3) Properties of geometric series:
Let's study the properties of geometric series together.
Through the above research, we find that there seems to be similarities between geometric series and arithmetic progression, which provides us with an idea to study the properties of geometric series: we can obtain the properties of geometric series by analogy with the properties of arithmetic progression.
Question 4: If {an} is a arithmetic progression, what is its nature?
(According to the actual situation of students, students can be guided to find the law through specific examples, such as:
3. Example integration:
Answer: 1458 or 128.
Example 2, positive geometric series {an}, a6? a 15+a9？ If a 12=30, then log 15a 12a3? a20 =_ 10 ____。
Example 3. Given a arithmetic progression: 2, 4, 6, 8, 10, 12, 14, 16, 2n, can we take some terms from this series to form a new series {cn} so that {cn} is the common ratio of 2?
(This question is open-ended, and there is no unique answer. For example, for {cn}: 2, 4, 8, 16, 2n, then ck=2k=2? 2k- 1, so the k-th item in {cn} is the 2k- 1 item in arithmetic progression. The key is to understand the general formula)
1, summary:
Today, we mainly studied the concept, general formula and properties of geometric series. Through today's study,
Not only did you learn geometric series, but more importantly, you learned the scientific thinking process of analogical conjecture proof.
2. Homework:
P 129: 1，2，3
Question: Take some items from arithmetic progression: 2, 4, 6, 8, 10, 12, 14, 16, 2n to form a new one.
Explain the design description:
1. Teaching objectives and difficulties: First of all, as the first lesson of geometric series, the concept, general formula and properties of geometric series are the basis for students to learn geometric series next, which must be implemented; Secondly, in addition to imparting knowledge, mathematics teaching is more important to impart scientific research methods. Geometric progression studied after geometric progression, so his study of geometric progression must be combined with his study of geometric progression. Through the analogy study of geometric series and arithmetic progression, it is beneficial to cultivate students' scientific research methods of analogy conjecture proof. This has become the focus of this class.
2. Instructional design process: This course mainly starts from the following aspects:
1) By reviewing the definition of arithmetic progression, the definition of geometric progression is obtained by analogy;
2) Derivation of the general formula of geometric series;
3) Properties of geometric series;
Consciously guide students to review arithmetic progression's definition and explore the general formula, on the one hand, make students review the old.
Knowledge, through association, enables students to lay the foundation for analogical exploration of the definition and general formula of geometric series.
After the definition of geometric series is obtained by analogy, several specific series are determined, aiming at? Special, general, special? The law of cognition enables students to experience the application of reasonable reasoning methods such as observation, analogy and induction. Cultivate students' ability to apply knowledge.
After getting the definition of geometric series, exploring the general formula of geometric series is another key point. Here, through the design of question 3, students have a psychological tendency to have to consider the general formula, resulting in cognitive conflicts among students, so that students can actively complete the acceptance of knowledge.
By comparing the general formulas of arithmetic progression and geometric progression, let the students know the similarity of arithmetic and ratio, and pave the way for the following analogy to learn the essence of geometric progression.
The research on the nature of equivalence ratio is the climax of this class. By analogy,
About the example design: It emphasizes the application of knowledge and is open, so that students can better grasp the content of this lesson.
Teaching Preparation of Teaching Plan II of Geometry Series, a compulsory course in senior high school mathematics.
Teaching objectives
Knowledge goal: to enable students to master the definition and general formula of geometric series, discover some simple properties of geometric series, and solve some practical problems by using the definition and general formula.
Ability goal: to cultivate the ability to find and solve problems by inductive analogy and the computing ability by using equation thought.
Moral education goal: cultivate positive learning style, enhance the application consciousness of mathematical concepts, and cultivate learning interest in personality quality.
Emphasis and difficulty in teaching
This section focuses on the definition, general formula and simple application of geometric series, and its solutions are induction and analogy.
The difficulty of this section is a deep understanding of the definition and general formula of geometric series. The key to breaking through the difficulties lies in adhering to the definition. In addition, it is also difficult to use definitions, formulas and properties flexibly to solve some related problems.
teaching process
Second, the analysis of teaching methods and learning methods
(1) through examples, let students find the law. Let students experience the formation and development of knowledge in the problem situation, and strive to make students learn to look at problems by analogy. (2) Create a democratic teaching atmosphere, grasp the emotional communication between teachers and students, let students participate in the whole teaching process, let students play a leading role, and the teacher is the director. ③ Strive for comprehensiveness and timeliness of feedback. Through well-designed questions, students' thinking can be mobilized, and teachers can make appropriate adjustments to the questions answered by students. (4) Give students time and space to think, don't throw the results to students in a hurry, let students observe, analyze and draw the results by analogy, and the teacher will make comments, so as to gradually develop a scientific and rigorous learning attitude and improve students' reasoning ability. ⑤ Take inspiring thinking as the core, moderately inspire, leave room, draw from it, and draw it. Doing so increases students' participation opportunities, enhances students' awareness of participation, teaches students how to acquire knowledge and think about problems, makes students truly become the main body of teaching, enables students to learn to learn, and improves students' interest and ability in learning.
Third, the teaching program design
(4) Arithmetic average: If A, A and B become arithmetic progression, then A is called the arithmetic average of A and B. ..
Description: By reviewing arithmetic progression's relevant knowledge, learn the content of this lesson by analogy, and use the familiar arithmetic progression content to disperse the difficulties of this lesson.
2. Introduce new courses
In the introduction of this chapter, the number of grains in each grid of the chessboard is:
1 , 2 , 4 , 8 , ? , 263
Let's look at two series:
5 , 25 , 125 , 625 , ...
?
Explanation: Guide students to pass? Observation, analysis and summary? , analogy arithmetic progression's definition of geometric series, in order to further understand the definition, the following questions are given:
Determine whether the following series is a geometric series. If yes, write the common ratio q; If not, give reasons and then answer the following questions.
- 1 , -2 , -4 , -8 ?
- 1 , 2 , -4 , 8 ?
- 1 , - 1 , - 1 , - 1 ?
1 , 0 , 1 , 0 ?
Ask the question: (1) Can the common ratio q be zero? Why? What about the first item a 1?
(2) What series is the common ratio q= 1?
(3)q & gt； Is 0 an increasing series? Q<0 is decreasing?
Description: Through the question and answer between teachers and students, the initiative and enthusiasm of students are fully mobilized, the classroom atmosphere is enlivened, and students' oral expression ability and improvisation ability are cultivated. In addition, through interesting questions, improve students' interest in learning. Stimulate students' strong desire to discover the definition and general formula of geometric series.
3. Try to deduce the general formula.
Ask the students to review the derivation process of arithmetic progression's general term formula and lead to the derivation of geometric progression's general term formula.
Deduction: iterative method.
Description: Students learn the first method from special to general methods, discover laws from the times, and cultivate students' observation ability; In addition, the characteristics of arithmetic progression are recalled and analogized into geometric series to cultivate students' analogical ability and the ability to transform new knowledge into old knowledge. The second method is to let students master it? By? This idea.
4. Explore the images of geometric series.
Arithmetic progression's image can be regarded as a set of isolated points on a straight line. What can you get by observing the general formula of geometric series? What is its image?
Variant 2. In the geometric series {an}, a2 = 2, a9 = 32, q 。
(Students answer by themselves. )
Note: The purpose of the example 1 is to familiarize students with the formula and apply it to practice. Example 2 and its variant are to make students understand that if they know any three of the four quantities A 1, Q, N and An in the formula, they can find another one. And from these questions, we can master the conventional elimination method in geometric series operation.
6. Explore the essence of geometric series
Compare the properties of arithmetic progression, guess the properties of geometric progression, and then guide the derivation.
7. Property application
Example 3. In the geometric series {an}, a5 = 2, a 10 = 10, and find a 15.
Let the students do it themselves and find various ways to solve the problem. )
Method 1: By solving the problem-meaning sequence equations.
Method 2: Use Attribute 2
Method 3: Use Attribute 3
Example 4 (see textbook example 3) It is known that the series {an} and {bn} are geometric series with the same number of terms. Verification: {an? Bn} is a geometric series.
8. Summary
In order to make students further organize and systematize what they have learned, and at the same time cultivate students' summing-up and cognitive ability after practice, teachers guide students to sum up this lesson.
1, the definition of geometric series, how to judge whether a series is geometric series?
2, the general formula of geometric series, the meaning of each letter.
3, geometric series should pay attention to those problems (a 1? 0，q？ 0)
4. Images of geometric series
5, the application of general formula (know three as one)
6. Properties of geometric series
7. The concept of geometric series (pay attention to two points) ① Only two numbers with the same sign have equal ratio terms.
(2) There are two proportional terms, and they are opposites)
8. Main ideas adopted in this course
Analogical thinking
9. Arrange homework
Exercise 3.4 1②, ④ 3. 8.9.
10. Blackboard design