1. Teaching content:
The principle of classified counting and step-by-step counting are also the focus of this lesson.
2. Conceptual analysis:
The principle of classified counting and the principle of step-by-step counting are both the principles of calculating how many different ways a thing can be done, also known as the principle of addition principle sum multiplication. The difference lies in: the precondition of using addition principle is that there are n methods to accomplish a thing, and any method in any method can be accomplished independently, that is to say, the methods to accomplish this thing are independent of each other, so the total number of methods is the sum of all methods; The premise of applying the principle of multiplication is that it takes n steps to complete one thing, and all the steps can be completed in turn. In other words, all the steps to complete this thing are interdependent, so the total number of methods is the product of the number of methods in each step.
3. The position and function of the two counting principles:
Classified counting principle and step-by-step counting principle are the basic laws summarized by people on the basis of a lot of practical experience, which embody two common methods of decomposing problems, namely, classified solution or step-by-step solution. This is not only the basis of deducing the formulas of permutation number and combination number in the future, but also the thinking and method of solving problems throughout this chapter.
Second, the teaching objectives:
1. Knowledge and skill goal: Understand and master the principles of classified counting and step-by-step counting, and use them to analyze and solve some simple application problems.
2. Process and Methods Purpose: To create situations and reduce some practical problems to a problem of classification or gradual counting, so as to improve students' constructive thinking ability; Use special to general views in the summary; Students can have a deeper understanding of the two counting principles by analogy and analogy when solving problems.
3. Emotional and attitudinal goals: Through group activities, cultivate students' good study habits of serious thinking and careful analysis, so that students can make correct analysis and accurate judgment when facing complex affairs and phenomena in real life, and then put forward a perfect treatment plan to understand the internal relationship and indivisibility between mathematical knowledge and real life. Let students feel the cordial and harmonious learning atmosphere, and further cultivate their awareness and ability of cooperation and communication in the activities.
Third, the analysis of students' learning situation:
1. Cognitive basic analysis:
By junior high school, students have learned to solve some counting problems with enumeration or tree diagram, and they have acquired certain induction and analogy ability, and can also solve some simple practical problems, forming a "recent development zone" for students' thinking.
2. Analysis of possible learning disabilities:
The premise of correctly using the two counting principles is that students should make clear the conditions for using the two counting principles: addition principle for classification and multiplication principle for step by step, which is easy for students to understand. Addition and multiplication can be learned in primary schools, so what's the difference between middle schools and previous ones? The difference is that the primary school stage focuses on the pursuit of operation results, while ignoring the deep-seated ideas contained in the process; The two principles profoundly reflect the most basic idea of human counting, that is, "decomposition". More specifically, it is to divide the number of ways to accomplish a thing into categories or steps. Students often have some obstacles in judging whether a thing is classified or completed step by step, and some students find it difficult to understand the operation result of multiplication principle. Therefore, the teaching difficulties of this lesson are defined as: (1) How to judge whether a thing is completed by classification or step by step; (2) Understand the operation method in step-by-step counting principle, that is, the total number of methods is the product of the number of methods in each step.
3. Breakthrough analysis:
To accurately judge whether a thing is classified or completed step by step, we must first make clear what it is and how to accomplish it. In the process of analysis, we will find that some things can be done independently by some methods, and some things need multiple steps to complete. Use classification if you can do it independently, and use step by step if you need to do it in multiple steps. Therefore, two group activities are designed for students to experience.
For the operation results of the step-by-step counting principle, we can make a breakthrough by using the tree diagram and combining the understanding of multiplication in primary schools.
Fourthly, the analysis of teaching strategies:
Most of the quotations and examples in this lesson can be understood by students through preview. In order to be close to students' real life and stimulate students' interest in learning, the familiar campus life cases are selected in the creation of situations and the selection of cases.
In this class, the teacher guides and inspires students to study in groups. Using multimedia to show the problem situation, let students analyze and compare in detail through group activities, and then summarize, which embodies the thinking process from special to general, not only pays attention to students' cognitive basis, but also urges students to acquire knowledge on the basis of original cognition, improve their thinking ability and maintain a high level of thinking activities, which is in line with students' cognitive laws.
Students' understanding of problems can be supplemented and improved in group cooperation and communication. Teachers can better understand students' understanding from students' answers and mutual evaluation among students.
Verb (abbreviation of verb) teaching process;
1. Create situations and reveal topics.
Students, we will move to the first teaching building to study next semester. Have you observed how many stairs there are in the first teaching building? (4)
Suppose the classroom in our class is on the second floor. How many different ways are there from the first floor to the second floor? (Ask a classmate to answer)
Suppose the classroom in our class is on the sixth floor. How many different ways are there to walk from the first floor to the sixth floor? Let the students fully discuss and get confused in the process of solving problems, thus stimulating students' thirst for knowledge. )
These problems are actually counting problems, and they are all about calculating how many different ways to accomplish a thing. The principle of classified counting and step-by-step counting that we are going to learn today is to solve this kind of problem.
Calculate how many different ways to accomplish a thing, and how to do it? It's like asking you to finish something. What's your first thought? What's this? ) Then what came to mind? (how to do it? In the process of analysis, we know how to complete it, and then count all the completed methods.
Today, our study starts from these two aspects.
Design purpose: Select the materials around students as examples of introducing new courses, stimulate students' learning accumulation with simple and familiar problem situations, and let students explore urgently.
2. Explore layer by layer and build new knowledge.
What should we accomplish in the first question just now? How to complete it?
From the first floor to the second floor: (choose any stairs) in one step, directly completed.
Second question, what should we accomplish? How to complete it? (First to the second floor, then to the third floor, ...)
From the first floor to the sixth floor: you can't do it directly, you need to do it step by step.
The first step: from the first floor to the second floor; Step 2: From the second floor to the third floor; Step 3: From the third floor to the fourth floor; Step 4: From the fourth floor to the fifth floor; Step 5: From the fifth floor to the sixth floor.
Comparing the completion process of two things, can you find their differences?
Complete one thing: one step at a time, directly; You can't do it directly, you need to do it step by step.
Study group activity 1: Discuss how to accomplish the following things.
(After the students discuss in their respective study groups, the group representatives will speak. )
Design purpose: let students feel that one thing can be solved by classification or step by step.
Situation 1. There are 4 boys and 8 girls among the candidates for program host.
(1) If you choose one person to host the program; (You can choose male students or female students, both of which are done directly. In your selection method, according to the gender of students, you can divide them into two categories, one is to choose boys and the other is to choose girls. Whether you choose a man or a woman, you can do it independently. )
(2) If you choose a male classmate and a female classmate to co-host the program. (you can't do it directly, it needs to be divided into two steps. The first step is to choose male students, and the second step is to choose female students. No matter whether you choose a male classmate or a female classmate, there is no step that you can't finish. The relationship between these two steps is interdependent. )
Scene 2: There are 40 different Chinese books, 30 different math books and 20 different English books on the shelf.
(1) Take a book from the shelf;
(2) Choose three books with different themes.
In the process of completion, we also found that what can be completed directly can often be completed according to certain standards; What can't be done directly needs to be done step by step.
When can the classification be completed and when should it be completed step by step?
Use classification if you can do it independently, and use step by step if you need to do it in multiple steps.
Study Group Activity 2: Calculate how many different ways to accomplish the following things.
Design purpose: let students use the existing knowledge to calculate, improve the method and find the law in the communication with classmates.
Classification completed:
(1) There are 4 boys and 8 girls among the candidates who host the program. If you choose one to host the program, how many different ways can * * * choose?
The first category: there are four ways to choose a man; Category II: There are eight ways to choose a woman, * * * 4+8 = 12.
There are 40 different Chinese books, 30 different math books and 20 different English books on the shelf. How many different ways are there to choose a book from the shelf?
The first category: there are 40 ways to choose words; The second category: there are 30 ways to select numbers; Category III: There are 20 ways to choose English, * * * 40+30+20 = 90.
Step by step:
(1) There are 4 boys and 8 girls among the candidates who host the program. How many different ways can * * * choose if a male classmate and a female classmate are chosen to host the program together?
1 of men and women have 8 choices, ***4 men; Women 1 There are four options for bringing a man, ***8 women; (tree view)
The first step: there are four ways to choose a man; Step 2: There are eight ways to choose a woman.
(8+8+8+8=4+4+4+4+4+4+4+4==32)
(2) There are 40 different Chinese books, 30 different math books and 20 different English books on the bookshelf. Choose three books on different topics. How many different ways to choose from?
Analysis: There are 1, 200 ways to complete the first and second steps (choose a Chinese book and a math book). When choosing an English book, each of the above methods corresponds to 20 methods, so there are 24,000 methods to choose from.
3. Comparative induction, deepen the concept
Study group activity 3: think about it: how to calculate the counting problem of classification completion?
How to calculate the counting problem completed step by step?
Design purpose: Students can find out the rules through the analysis and calculation of specific cases, express them in their own language, and exercise their generalization ability.
Classified solution: the number of all methods to accomplish this is the sum of the number of various methods.
Step-by-step solution: all the ways to accomplish this are the products of each step.
(Two students make concluding remarks) This is the content of the principle of classified counting and the principle of step-by-step counting that we are going to learn today. (Computer display, students read aloud)
Principle of classification and counting: there are n ways to complete a thing, 1 way has different ways, the second way has different ways ... N ways have different ways, so the way to complete it is also different.
Step-by-step counting principle: to complete one thing, it needs to be divided into n steps. Step 1, Step 2 has different methods ... Step N, so there are different ways to complete it.
Classification counting principle is also called addition principle, and step-by-step counting principle is also called multiplication principle. Can you tell me why they are named? (The result is added by various method numbers, and the result is multiplied by the method number of each step.)
When to use addition principle and when to use the principle of multiplication?
Again, every method in addition principle can be done independently, but every step of the multiplication principle cannot be done independently.
4. Apply what you have learned and cultivate your ability
We have learned two counting principles. How can we use them to solve the problem of climbing stairs?
From the first floor to the second floor: the classification is completed, and there are * * * four categories, each with 1 move, so * * * four different moves.
From the first floor to the sixth floor: step by step, * * * five steps, each 1 step has four moves, so * * * has different moves.
Design purpose: echo before and after, solve problems with what you have learned, and achieve the purpose of applying what you have learned.
5. Summing up reflection and raising awareness
What kind of gains do students have in knowledge through the study of this course?
(1) knowledge accumulation: classification counting principle and step-by-step counting principle.
Students, think again, how did we get these two counting principles?
(2) Thinking experience: Through some specific problems, the principles of classified counting and step-by-step counting are summarized respectively. From special to general is one of the important ways of thinking.
Teacher: What is achievement and how to achieve it? Classification or step by step? Be sure to distinguish! Classification is independent, and classification is additive. Steps must depend on each other, and steps are multiplied.
The principle of classified counting and the principle of step-by-step counting embody two common methods of decomposing problems, namely, classified solving or step-by-step solving. This is not only the basis of deducing the formulas of permutation number and combination number in the future, but also the thinking and method of solving problems throughout this chapter.
6. Assign homework and expand your knowledge.
Exercise: There are five letters in one pocket and four letters in the other. The content of each letter is different.
(1) How many different ways are there to take out 1 letters from two pockets?
(2) How many different ways are there to take out 1 letter from two pockets?
(3) How many different ways are there to put nine letters in these two pockets into four mailboxes respectively?