Arithmetic progression lesson plan Fan Wenyi
Teaching objectives
Knowledge and skill goal: understand arithmetic progression's definition; Will find the value of an item according to arithmetic progression's general formula; According to the first few terms of arithmetic progression, we can find the general term formula of series.
Process and Method Objective: To improve students' ability to think and solve problems through inspiration, discussion, guidance, practice while teaching and feedback.
The goal of emotion, attitude and values: to cultivate students' logical reasoning ability; Cultivate students' spirit of learning knowledge through exploration and enhance their awareness of mutual cooperation and exchange.
Teaching emphasis: arithmetic progression's general formula will be sought.
Teaching difficulty: the derivation of arithmetic progression's general formula.
Teaching preparation: courseware
Teaching process:
First, create situations and introduce topics.
As shown in figure 1: the bottom of the V-shaped shelf for stacking pencils.
Put 1 pencil on the first floor, and put more 1 pencil on each floor above than on each floor below.
Yes, the pencils in this V-shaped frame are arranged from the bottom to the top.
The number of pencils constitutes a series: 1, 2, 3, 4, ...
(2) A cinema has 20 rows of seats, and the number of seats in each row of this cinema is from 1 row to form a series:
38,40,42,44,46,……
③ In the national uniform shoe size, the sizes of adult women's shoes (indicating the sole length in centimeters) are arranged as follows: 25, 24.5, 24, 23.5, 23, 22.5, 22, 2 1.5.
Interaction between teachers and students, exploring new knowledge
Teacher: Please observe carefully. What changes have you found in these three series?
Health: series ① starts from the second item, and the difference between each item and the previous item is equal to;
Sequence ② starts from the second item, and the difference between each item and its previous item is equal to;
Sequence ③ starts from the second item, and the difference between each item and its previous item is equal to;
[Design Description: Teaching while giving feedback will help teachers keep abreast of students' understanding of new knowledge and enhance students' confidence in learning mathematics well]
Teachers guide students to observe the characteristics of the above sequences ①, ② and ③.
Question 1: What are the * * * characteristics of the above three series?
Student: Starting from the second item, the difference between each item and the previous item is equal to the same constant.
Teacher: So we got the definition of arithmetic progression.
& ltI > arithmetic progression's definition: If the difference between each term of a series from its second term to its previous term is equal to the same constant, then the series is called arithmetic progression; This constant is called arithmetic progression's tolerance, usually expressed by the letter D, and the mathematical expression of arithmetic progression's tolerance D is:
Basic training: 1, the tolerance of the above sequence d 1 =; Tolerance d of sequence ② =;
Tolerance d of sequence ③ =
[Design Description: Help students clear the obstacles of language and symbol conversion]
2. Which of the following series is arithmetic progression? If so, find its tolerance; If not, explain why.
6, 10, 14, 18,22,……; (2)9,8,7,6,5,4,3,2; (3)3,3,3,3,3,3; (4) 1,0, 1,0, 1,0, 1,0.
Question 2: Does any series have to be arithmetic progression? If it is arithmetic progression, must the tolerance be positive?
Teachers and students discuss and draw a conclusion:
A arithmetic progression must have the following characteristics: starting from the second term, the difference between each term and its previous term is equal to the same constant;
(2) The tolerance d of arithmetic progression may be positive, negative or zero.
[Design Description: Starting from the specific series, it is helpful for students with more basic differences to understand the definition of arithmetic numbers and judge whether the series is transformed into arithmetic progression. The specific steps are: find the difference between the last one and the previous one to see if these differences are equal. ]
Question 3: The mathematical expression of arithmetic progression's tolerance d is,
What formulas can be used to find the tolerance d?
Teacher-student activities, etc.
Variant:
Question 4: If arithmetic progression only knows the first tolerance D, how to express the other terms of this series?
Teachers and students * * * and activities:
…,
[Design Description: Questions 3 and 4 are put forward to train students' deformation thinking and recursive thinking, thus deriving arithmetic progression's general formula and deformation formula that students can easily understand].
& lt seconds > General formula of arithmetic progression:
Arithmetic progression's lesson plan Fan Wener
Arithmetic progression's teaching plan design
Teachers give lessons to students. Topic 3.2. 1 arithmetic progression (I) Definition of the new teaching goal knowledge goal arithmetic progression.
Arithmetic progression's general formula. Arithmetic progression's definition is clear about the goal of ability.
Master arithmetic progression's general formula and use it to solve problems, and cultivate students' observation ability with emotional goals.
Further improve students' reasoning and inductive ability.
Cultivate students' application consciousness. Teaching focuses on understanding and mastering arithmetic progression's definition.
Derivation and application of arithmetic progression's general formula. Understanding, Mastering and Application of arithmetic progression's "Arithmetic" Characteristics. Review of the design intention of teaching links and teaching contents in the teaching process (2 minutes)
Definition of sequence, general term formula and recurrence formula of sequence.
Introduction (3 minutes)
Someone wants to decorate the Christmas tree with colored lights. This man likes to follow certain rules. He installed 65,438+0 colored lights on the top of the Christmas tree, including 4 on the first floor, 7 on the second floor, 65,438+00 on the third floor and 65,438+03 on the fourth floor. If there is a fifth floor, how many colored lights do you think he will install? What are his rules?
Can you fill in the appropriate number in () according to the law?
( 1) 1, 4, 7, 10, 13,( )
(2)2 1, 2 1.5, 22, ( ), 23, 23.5,…
(3)8,( ), 2, - 1, -4, …
(4)-7, - 1 1, - 15, ( ), -23
* * * Same feature: From the second term, the difference between each term and its previous term is equal to the same constant. This series is called arithmetic progression.
Teach a new lesson (16 minutes)
1. Definition of arithmetic progression: Generally speaking, if a series starts from the second term and the difference between each term and its 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..
Symbol:
Teacher's activities: analyze the definition, emphasize the key points, and help students understand and master.
Question: 1. What is the tolerance of (1)(2)(3)(4) series?
2.(5) 1, 3, 5, 7, 9, 2, 4, 6, 8, 10
(6)5, 5, 5, 5, 5, 5 ... Is it arithmetic progression?
3. Find arithmetic progression 1, 4, 7, 10, 13, 16,.
Teachers and students discuss and answer together.
Second, arithmetic progression's general formula
If the first term of arithmetic progression is tolerance D, according to its definition, we can get:
Namely:
Namely:
Namely:
Thus, arithmetic progression's general formula can be summarized as follows:
∴ If a series is known as arithmetic progression, as long as we know the first term and the tolerance d, we can find the general term.
Thinking: Given arithmetic progression's m-th term, tolerance D, what is the general formula of arithmetic progression? A:
Example description (8 minutes)
Arithmetic progression's teaching plan, Fan Wensan.
First, the analysis of teaching content
This lesson is the first lesson of arithmetic progression, the second chapter of Mathematics 5 series, a standard experimental textbook for senior high school (People's Education Edition).
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 the thinking methods of "association" and "analogy" for studying geometric progression in the future.
Second, the analysis of students' learning situation
The teaching content is aimed at senior two students. After one year's study in senior high school, most students are rich in knowledge and experience, and have strong abstract thinking ability and deductive reasoning ability. But some students may have a weak foundation. Therefore, in teaching, we should start from concrete life cases, stimulate students' interest in learning, and pay attention to guiding and inspiring students to actively study mathematics, so as to further improve students' thinking ability.
Third, the design ideas
1. Teaching methods
⑴ Inductive thinking method: This method is helpful for students to actively construct knowledge; Conducive to highlighting key points and breaking through difficulties; It is conducive to mobilizing students' initiative and enthusiasm and giving play to their creativity.
⑵ Group discussion: It is beneficial for students to communicate, find and solve problems in time, and arouse students' enthusiasm.
⑶ Emphasize the combination with practice: we can consolidate what we have learned in time, grasp the key points and break through the difficulties.
Study law
Guide students to summarize the characteristics of array from four practical problems (counting, setting women's weightlifting prize, reservoir water level and saving) and abstract the concept of arithmetic progression; Then, according to the characteristics of arithmetic progression's concept, the general formula of arithmetic progression is deduced. It can guide students of all abilities to understand the multiple deductive thinking method.
Arithmetic progression's general formula is deduced by various methods.
When guiding the analysis, leave a "blank" for students to associate and explore, and encourage students to question boldly, express their opinions around the center, and clarify the thinking methods and problems to be solved.
Fourth, teaching objectives.
Through the study of this lesson, students can understand and master arithmetic progression's concept, judge whether a series is arithmetic progression according to the definition, guide students to understand the derivation process and thought of arithmetic progression's general formula, master arithmetic progression's general formula and the first N sum formula, and solve simple practical problems; In this process, students' abilities of observation, analysis, induction and reasoning are cultivated. 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.
Key points and difficulties in teaching verbs (abbreviation of verb)
Key points:
(1) the concept of arithmetic progression.
② The derivation process and application of arithmetic progression's general term formula.
Difficulties:
(1) to understand arithmetic progression's arithmetic characteristics and the meaning of the general formula.
② Understanding that arithmetic progression is a functional model.
Key:
Understanding of arithmetic progression's concept and the resulting "natural" method.
Sixth, the teaching process.
Situational Design of Teaching Links and Design of Students' Activity Intention Creating Scenes In the Zhang Qiujian Su 'an Sutra in the Northern and Southern Dynasties, there is a topic "Today, there are ten people, each with one person, and the palace gives them a grade difference. The top three people entered first, and they got four catties of gold. Hold on, the bottom four people got three catties of gold, and the middle three people were given more according to the grade. Ask each gold geometry if they have arrived. "
How to solve this problem? Listen to the class, introduce exploratory research, students observe and analyze, and get the answer;
In real life, we often count this way, starting from 0 and counting every five, and we can get the sequence: 0, 5, _ _ _ _ _ _, _ _ _ _ _, _ _ _ _ _, ...
In order to ensure a good living environment for high-quality fish, the managers of the reservoir regularly release water to clean up the miscellaneous fish in the reservoir. If the water level of the reservoir is 18cm, the daily water level of natural water discharge will be reduced by 2.5m, and the lowest is 5m ... Then, from the beginning of water discharge to the day when cleaning can be carried out, the daily water level of the reservoir consists of a series (unit: m): 18,15,1.
0,5, 10, 15,20,…… ①
18, 15.5, 13, 10.5,8,5.5 ②
See what * * * has in common with these series? Observation and analysis give the answer:
Guide students to observe the relationship between two adjacent projects and draw the following conclusions:
For sequence ①, starting from the second item, the difference between each item and the previous item is equal to 5;
For sequence ②, the difference between each term and the previous term is equal to-2.5 from the second term;
The students come to the conclusion that the difference between each item in the above two series and the previous item is equal to the same constant from the second item (that is, each item has the characteristic that the difference between two adjacent items is the same constant). Through analysis, stimulate students' interest in learning and exploring knowledge, and guide to reveal the * * * characteristics of sequence. Summarize and improve [arithmetic progression's concept]
We call these series arithmetic progression. Please try to define arithmetic progression according to the characteristics of arithmetic progression we just analyzed:
Arithmetic progression: Generally speaking, if a series starts from the second term, and the difference between each term and its previous term is equal to the same constant, then the series is called arithmetic progression.
This constant is called arithmetic progression's tolerance, which is usually represented by the letter D, so for the above two groups of arithmetic progression, their tolerances are 5, 5 and -2.5 in turn. Students carefully read the related concepts in the textbook and find out the key words. Find out the key words through students' own reading materials, improve their reading level and thinking generalization ability, and learn to grasp the key points. Question: If a number A is inserted between and to make A a arithmetic progression, what conditions should A meet? Answer: Because A, A and B make up a arithmetic progression, we can know that A-A = B-A from the definition.
Therefore, it is necessary for students to participate in the formation of knowledge and gain a sense of accomplishment in mathematics learning. Arithmetic progression, which consists of three numbers A, A and B, can be regarded as the simplest arithmetic progression. At this time, a is called the arithmetic average of a and B.
It is not difficult to find that in a arithmetic progression, starting from the second term, every term (except the last term with finite series) is the arithmetic average of the previous term and the latter term.
For example, in the sequence: 1, 3, 5, 7, 9, 1, 13… where 5 is the arithmetic average of 3 and 7,1and 9.
9 is the arithmetic mean of 7 and 1 1, and the arithmetic mean of 5 and 13.
It seems that,
Therefore, in a arithmetic progression, if m+n=p+q,
Then explore deeply, draw more general conclusions, guide learning to explore deeply, and improve students' learning level. Summarize and improve [arithmetic progression's general formula]
For the above arithmetic progression, can it be expressed by a general formula? This is what we will learn next.
(1) By studying the relationship between the nth item of the series and the serial number n, write the general term formula of the series ... Now students write the general term formulas of these three groups of arithmetic progression according to the definition of the general term formula. Students write the general formula after analysis:
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