Teaching Design of Mathematics arithmetic progression Part I: Teaching Objectives
1. Knowledge and skills
(1) Understand the definition of arithmetic progression, and use the definition to judge whether a series is arithmetic progression;
(2) The general formula of accounting arithmetic progression and its derivation process:
(3) arithmetic progression's general formula will be applied to solve simple problems.
2. Process and method
In the process of understanding the definition, derivation and application of general formula, students' abilities of observation, analysis and induction and strict logical thinking are cultivated, and the cognitive laws from special to general and from general to special are experienced, so that the ability of being familiar with guessing and induction is improved, and the ideas of functions and equations are infiltrated.
3. Emotions, attitudes and values
Through students' autonomous learning, mutual communication and exploration activities under the guidance of teachers, students' spirit of seeking knowledge for active exploration and discovery is cultivated, students' interest in learning is stimulated, and students can feel the joy of success. In the process of solving problems, make students form the good habit of careful observation, careful analysis and good at summing up.
Teaching focus
(1) the concept of arithmetic progression; ② arithmetic progression's general formula.
Teaching difficulties
1) Know about arithmetic progression? Equal difference? The characteristics and meaning of the general formula; ② The derivation process of arithmetic progression's general term formula.
Analysis of learning situation
The students I teach are students in Class 7, Grade 1 (students in parallel classes). After one year's mathematics study in senior high school, most students already have rich knowledge and experience, and their intelligence has reached the stage of formal operation, with strong abstract thinking ability and deductive reasoning ability. However, some students have a weak foundation and are not very interested in learning mathematics. Therefore, I pay attention to guidance and inspiration in teaching, and pay attention to specific life examples.
Design concept
1. Teaching methods
① Enlightening and guiding 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.
Group discussion: it is beneficial for students to communicate, find and solve problems in time and arouse their enthusiasm.
③ Combination of stress and practice: It can consolidate the learned content in time, grasp the key points and break through the difficulties.
Study law
Guide students to summarize the characteristics of array from three practical problems (counting problem, reservoir water level problem and saving problem) 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. Can guide students of all abilities to understand the multiple deductive thinking method.
teaching process
First, create situations and introduce new lessons.
1. Starting from 0, arrange multiples of 5 in descending order. What's the order?
2. In order to ensure a good living environment for high-quality fish, reservoir managers clean up the miscellaneous fish in the reservoir by regularly releasing water. If the water level of a reservoir is 18m, the daily water level will drop by 2.5m, with a minimum of 5m. What sequence does the daily water level (unit: m) form from the day when the reservoir is discharged to the day when it can be cleaned?
3. China's current savings system stipulates that the way for banks to pay deposit interest is simple interest, that is, the principal and interest are calculated without interest. The formula for calculating the sum of principal and interest according to simple interest is: sum of principal and interest = principal? (1+ interest rate? Term). If the current deposit is 10000 yuan and the annual interest rate is 0.72%, what series does the sum of principal and interest at the end of each year (unit: yuan) form according to simple interest calculation?
Teacher: The figures in the above three questions contain three columns of figures.
Student:
1:0,5, 10, 15,20,25,? .
2: 18, 15.5, 13, 10.5,8,5.5.
3: 10072, 10 144, 102 16, 10288, 10360.
(Intention: The introduction of examples is essentially to give the realistic background of arithmetic progression, and the purpose is to make students feel that arithmetic progression is a mathematical model that exists in real life. Through analysis, from the special to the general, it can stimulate students' autonomy in learning and exploring knowledge and cultivate their inductive ability.
Second, observation and induction form a definition.
①0,5, 10, 15,20,25,? .
② 18, 15.5, 13, 10.5,8,5.5.
③ 10072, 10 144, 102 16, 10288, 10360.
What are the same characteristics of the series 1 above?
Thinking 2 According to the * * * characteristics of the upper sequence, can you give a general definition of arithmetic progression?
Thinking 3: Can the above written language be converted into mathematical symbol language?
Teacher: Guide the students to think about the identity characteristics of these three sequences, then let the students master the characteristics of the sequence and sum up the concept of arithmetic progression.
Student: In group discussion, there may be different answers: the difference between the first number and the last number conforms to a certain law; These figures are all arranged in a certain order? As long as it is reasonable, teachers should give affirmation.
Teacher guidance summary: arithmetic progression's definition; In addition, teachers guide students to understand arithmetic progression's definition from the perspective of mathematical symbols.
(Design intent: through the observation and analysis of a certain number of perceptual materials, extract the essential attributes of perceptual materials; Make students realize the laws and characteristics of arithmetic progression; Let's catch it first: from the second term, the difference between each term and the previous term is the same constant? To realize the accurate expression of arithmetic progression's concept. )
Three: draw inferences from others and consolidate the definition.
1. Determine whether the following series is arithmetic progression? If yes, indicate the tolerance d.
( 1) 1, 1, 1, 1, 1;
(2) 1,0, 1,0, 1;
(3)2, 1,0,- 1,-2;
(4)4,7, 10, 13, 16.
The teacher shows the questions and the students think and answer them. The teacher corrected and emphasized the problems that should be paid attention to when seeking tolerance.
Note: The tolerance d is the difference between each item (from the second item) and its previous item to prevent the minuend from inverting from the minuend. The tolerance can be positive, negative or 0.
Design intention: to enhance students' understanding of arithmetic progression? Equal difference? Understanding and application of features).
2 Thinking 4: Let the general formula {an} of series be an=3n+ 1. Is this series arithmetic progression? Why?
(Design intention: strengthen arithmetic progression's proof definition method)
Fourthly, the general term is deduced by definition.
1. Known arithmetic progression: 8, 5, 2,? What about item 200?
2. Given that the first term of a arithmetic progression {an} is a 1 and the tolerance is d, how to find its arbitrary term an?
Teachers show questions, let students explore freely, and then choose representative performances or projection shows. According to the specific situation of students in the classroom, make specific evaluation and guidance, summarize the derivation methods, experience inductive thinking and accumulate general terms; Let the students try the common methods to deal with the problem of sequence.
Design intention: guide students to observe, summarize and guess, and cultivate students' reasonable reasoning ability. Students may find a variety of different solutions in the process of group cooperative inquiry. Teachers should comment on them one by one, and affirm and praise students' good brains and innovative quality in time, stimulate students' creative consciousness, encourage students to answer independently, and cultivate students' computing ability. )
Five: apply general terms to solve problems
1 judge whether 100 is arithmetic progression 2,9, 16. Articles? If yes, which one is it?
2 In arithmetic progression {an}, a5= 10, a 12=3 1. Find a 1, d and an.
Arithmetic progression 3,7, 1 1? Item 4 and item 10 of.
Teacher: Give the questions and let the students practice by themselves. The teacher checked the students' answers.
Student: The teacher asked the student representatives to sum up the thinking of solving such problems. Teacher added: If you know the first term and tolerance of arithmetic progression, you can get the general term formula.
(Design intention: It is mainly to be familiar with formulas and let students understand the relationship between formulas and equations. Preliminary understanding? Basic quantity method? Solve the arithmetic progression problem. )
Six: feedback exercises: textbook 13 page exercises 1
Seven: summary:
1. A definition:
Arithmetic progression's definition and definition expression.
2. A formula:
Arithmetic progression's general formula
3. Two applications:
Definition and application of general formula
Teacher: Let the students think about organization, find a few representatives to speak, and finally the teacher will make a supplement.
(Design intention: Guide students to think of all aspects involved in this lesson and communicate their relationships, so that students can re-understand and master the basic concepts at a new height and use them flexibly. )
Design reflection
This design is introduced from the series model in life, which is helpful to give full play to students' initiative in learning and enhance their interest in learning series. In the process of exploration, students summarize arithmetic progression's definition through analysis and observation, and then deduce the general formula from the definition, which strengthens the thinking process from concrete to abstract and from special to general, and helps to improve students' ability to analyze and solve problems. Heuristic method is adopted in this course, teachers ask questions and students explore and solve problems.
Teaching Design of Mathematics arithmetic progression Part II: [Teaching Objectives]
1. Knowledge and skills goal: master the concept of arithmetic progression; Understand the derivation process of arithmetic progression's general term formula; Understand the functional characteristics of arithmetic progression; Arithmetic progression's general formula can be used to solve some corresponding problems.
2. process and method objective: let students experience it personally? From the special point of view, the nature of the research object gradually expands to the general? This research process has cultivated their abilities of observation, analysis, induction and reasoning. Through step-by-step intensive exercises, students' ability to analyze and solve problems is cultivated.
3. Emotion, attitude and value goal: through arithmetic progression's study, cultivate students' spirit of active exploration and courage to discover; Make students gradually develop the good habit of careful observation, careful analysis and timely summary.
Sense of [teaching emphasis and difficulty]
1. Teaching emphasis: Understanding arithmetic progression's concept, derivation and application of general formula.
2. Teaching difficulties: (1) What are the difficulties in arithmetic progression? Equal difference? Master two words;
(2) Derivation of arithmetic progression's general term formula.
[Teaching process]
I. Presentation
Create a situation to introduce a topic: (In this lesson, we will learn a special series. Let's look at some examples. )
(1), in the past three hundred years, people have observed Halley's comet at the following times:
1682, 1758, 1834, 19 10, 1986,()
Can you predict the approximate time of the next observation of Halley's comet? What is the basis of judgment?
(2) In general, from the ground to the altitude of 1 1km, the temperature changes with altitude, which conforms to certain laws. Please estimate the temperature at the summit of Mount Everest according to the table below.
Thinking: Fill in (3) and (4) according to the previous rules:
(3) 1,4,7, 10,(), 16,?
(4)2,0,-2,-4,-6,(),?
What are their similar laws?
Starting from the second term, the difference between each term and the previous term is equal to the same constant.
We call a series with this characteristic arithmetic progression.
Second, the new curriculum exploration
(A) the definition of arithmetic progression
1, the definition of arithmetic progression.
If a series starts from the second term and the difference between each term and the previous term is equal to the same constant, then the series is called arithmetic progression. This constant is called arithmetic progression's tolerance and is usually represented by the letter D. ..
What are the key words in the definition of (1)?
(2) What is the difference between the two numbers of tolerance d?
2. Mathematical expressions defined by arithmetic series:
Try it: Are they arithmetic progression?
( 1) 1,3,5,7,9,2,4,6,8, 10?
(2)5,5,5,5,5,5,?
(3)- 1,-3,-5,-7,-9,?
(4) sequence {an}, if an+ 1-an=3.
3. Definition of arithmetic median
Between the following two numbers, insert a number, and these three numbers become arithmetic progression:
( 1)、2、()、4(2)、- 12、()、0(3)a、()、b
If a number A is inserted between A and B to make A, A and B arithmetic progression, then A is called the arithmetic average of A and B..
(B) arithmetic progression's general formula
Explore 1: arithmetic progression's general formula (solution 1)
If arithmetic progression's first item is, tolerance is, then how to express this arithmetic progression? And then what?
According to arithmetic progression's definition:
So:
This shows that,
Therefore, arithmetic progression's general formula is,
Inquiry 2: arithmetic progression's General Formula (Solution 2)
According to arithmetic progression's definition:
Add the above expression-1 to get the general formula of arithmetic progression:
Third, application and exploration
Example 1, (1) Find arithmetic progression 8,5,2,? , item 20.
(2) What is the first item of arithmetic progression-5,9, 13? 40 1?
(2) Analysis: The key to judge whether -40 1 is a term of the sequence is to find the general term formula and judge whether there is a positive integer n, which essentially requires a positive integer solution of the equation.
Example 2. In arithmetic progression, it is known that = 10, =3 1, and the first term and the tolerance d are found.
Solution: by, get.
In the process of applying arithmetic progression's general formula an=a 1+(n- 1)d, for four variables an, a 1, n, d, we only need to know three of them to find the residual quantity, which is the idea of an equation.
Consolidation exercise
The first three items of 1. arithmetic progression {an} are a-6, -3a-5,-10a- 1, so a= ().
A. 1B。 - 1C。 -2D.2
2. A ladder, the first level is 33cm wide, the lowest level is 1 10cm wide, and there are 10 levels in the middle, and the width of each level is arithmetic progression. Find the tolerance D.
Four. abstract
1. arithmetic progression's general formula:
Tolerance;
2. arithmetic progression's calculation problem, usually know three of them, we can use the general formula an=a 1+(n- 1)d to find the remaining one;
3. To judge whether a series is a arithmetic progression, just look at whether it is a constant;
4. Use the thinking from special to general to discover the laws of the department of mathematics or solve mathematical problems.
Verb (short for verb) Homework:
1, required questions: Exercise 2.2, 1, 3,5 on page 40 of the textbook.
2, choose to do the problem: how to find the fastest speed: 1+2+3+? + 100=