Teaching design article 1 teaching content of sixth grade mathematics "pigeon cage principle";
The sixth grade mathematics volume has 70 pages, 7 1 page, examples 1, and examples 2.
Teaching objectives:
1, understand the general form of "pigeon hole principle".
2. Experience the inquiry process of "Pigeon Hole Principle", experience the learning methods of comparison and reasoning, and use "Pigeon Hole Principle" to solve simple practical problems.
4. Feel the charm of mathematics, improve learning interest and cultivate students' inquiry spirit.
Teaching focus:
Through the process of "pigeon coop principle", I have a preliminary understanding of "pigeon coop principle".
Teaching difficulties:
Understand the general law of "pigeon cage principle".
Teaching preparation:
A corresponding number of cups, pencils and courseware.
Teaching process:
First, the scene is introduced.
Let five students sit on four chairs at the same time, and draw the conclusion that no matter how they sit, there are always at least two students sitting on one chair.
Teacher: Students, do you want to know why? Today, we study a new and interesting math problem together.
Second, explore new knowledge.
1. Explore the problem of putting three pencils in two cups.
Teacher: Now put three pencils in two cups. How to put it? How many ways are there? Let's have a look around. What did you find?
After the exhibition, the students reported that the teacher wrote the corresponding blackboard book (3,0) (2, 1) to guide the students to observe, understand and say: No matter how you put it, there is always a cup with at least two pencils in it.
2. Teaching examples 1
(1) Teacher: How can you put four pencils in three cups if you push on like this? Will there be this conclusion? Let the students operate, take notes and observe carefully to see what they find.
(2) Students report the results and explain them in combination with the operation of learning tools. Teachers should make corresponding records.
(4,0,0) (3, 1,0) (2,2,0) (2, 1, 1)
It is not difficult for students to find the same conclusion as the previous question through operational observation. )
(3) Ask the students to read the dialog box in Example 1 after answering: No matter how you put it, there are always at least two pencils in a cup.
Teacher: What do you mean by "always"? What "at least" makes students understand what they mean.
Teacher: How can I always have the least pencils in a cup? Guide students to understand the necessity of "equal play".
The teacher shows the courseware demonstration, so that students can further understand "average play"
3. Explore the problem of n+ 1 pencil in n cups.
Teacher: Then let's think further. Six pencils are put in five cups. What do you think the conclusion will be?
Let the students think and find that there are always at least two pencils in a cup no matter how you put them.
Teacher: Seven pencils were put into six cups. What did you find?
……
After the students finished, the teacher asked: Is there always at least two pencils in a cup as long as the number of pencils is more than the number of cups 1? Let the students discuss and report in groups.
After the students report, guide them to test their ideas with experiments.
Teacher: Put 10 stick in 9 cups. How many sticks are there in a cup? (2 pieces)
Teacher: Put 100 stick in 99 cups. What is the conclusion? (2 pieces)
Step 4 summarize the rules
Teacher: Just now we all learned that the number of pencils is more than the number of cups 1, and the remainder is exactly 1. What if there are 2, 3 or 4 more pencils left than cups? What will be the conclusion?
(1) Explore putting five pencils in three cups. Anyway, how many pencils are there in a cup? Why?
A, first put it at the same table and say it again.
B, how to divide it?
After the students report, the teacher demonstrates that the five pens are divided into three cups on average. What about the other two? Can I put the remaining two in either cup? How to ensure at least?
Guide the students to know how to divide two pencils equally and put them in two cups respectively.
(2) Explore the conclusion of putting 15 pencil into four cups.
(3) Guide the students to the conclusion that there is always at least one cup in Shangjia 1.
(4) Teaching Example 2
Courseware demonstration:
1. Put five books in two drawers. No matter how you put it, there are always at least a few books in a drawer.
Put seven books in two drawers. No matter how you put it, there are always at least a few books in a drawer.
Put nine books in two drawers. No matter how you put it, there are always at least a few books in a drawer.
Student report
Summary: Anyway, there is always a drawer with at least "Shangjia 1" books.
Teacher: This is the interesting Pigeon Hole Principle, also called Pigeon Cage Principle, which was first put forward by German mathematician Dirichlet in the19th century, so it is also called Dirichlet Principle. This principle is widely used to solve practical problems. The application of "pigeon hole principle" is ever-changing. It can solve many interesting problems and often get some amazing results.
Third, solve the problem.
1 and 7 pens into 5 pen holders. No matter how you put it, there are always at least two pens in a pen container. Why?
2. Eight pigeons fly back to three pigeon coops. No matter how you fly, there are always at least three pigeons in a dovecote. Why?
Teacher: Finally, let's play another game. Have you all played poker? A * * * has several cards (54), and several cards (52) are reserved for King and Xiao Wang. The teacher asked a classmate to draw five cards at will. Without looking at it, the teacher knows that no matter how you draw, at least two cards are of the same suit. Is the teacher right? Why?
Fourth, class summary.
Blackboard design:
drawer principle
Number of pencils (number of objects) Number of cups (number of drawers) There is always a cup (drawer) with at least the number of objects in it.
3 2 2
4 3 2
6 5 2
7 6 2
100 99 2
n+ 1 n 2
5 3 5÷3= 1…2 1+ 1
15 4 15÷4=3…3 3+ 1
There is always a drawer with at least the number of objects: quotient+1.
Teaching Design of the Open Course of "Pigeon Cage Principle" in Grade 6 Mathematics 2 teaching material analysis
The understanding of pigeon hole principle is the content of the fifth chapter of the sixth grade mathematics book of People's Education Press. There is a kind of problem related to "existence" in mathematical problems. In this kind of problem, it is only necessary to determine the existence of an object (or person), without pointing out which object (or person) it is, and without explaining how to find out the existing object (or person). This kind of problem is based on what we call the "pigeon cage principle". The pigeon hole principle was first used by Dirichlet, a German mathematician in the19th century, to solve mathematical problems, so it is also called Dirichlet principle and pigeon's nest principle. 、
Analysis of learning situation
In this class, based on the idea that teachers are organizers, guides and collaborators, I created a new teaching structure with students participating in activities as the main line. Through several intuitive examples, the "pigeon hole principle" is introduced to students by hypothetical method, which is difficult for students to understand and feels abstract. In teaching, I combine the actual situation of our class, and use straws and cups that students are familiar with throughout the class, so that students can really know and understand the "pigeon hole principle" through hands-on operation in the activities, which is simple and easy for students to accept.
Teaching objectives
1. After going through the inquiry process of "pigeon hole principle" and having a preliminary understanding of "pigeon hole principle", we will use "pigeon hole principle" to solve simple practical problems.
2. The analogical ability developed by operation forms abstract mathematical thinking.
3. Feel the charm of mathematics through the flexible application of pigeon hole principle.
Teaching emphases and difficulties
Teaching focus
Through the exploration process of "pigeon coop principle", I have a preliminary understanding of "pigeon coop principle".
Teaching difficulties
Understand the "pigeon cage principle" and "simulate" some simple practical problems.
Teaching design for the open class of "Pigeon Cage Principle" in sixth grade mathematics 3 Teaching contents;
People's education printing plate sixth grade volume unit 5 mathematics wide angle
Teaching objectives:
1, a preliminary understanding of "pigeon hole principle".
2. Guide students to explore the general law of "pigeon hole principle" by operating enumeration or hypothesis.
3. Can use the pigeon hole principle to solve simple practical problems.
4. Through the concrete and abstract inquiry process, we can initially understand the pigeon hole principle, improve students' ability of orderly thinking and reasoning, and experience comparative learning methods.
Teaching emphasis: understanding and simple application of pigeon cage principle.
Teaching difficulty: find out the internal relationship between practical problems and pigeon coop principle.
Teaching process:
First, develop small games and introduce new lessons.
Teacher: Before class, let's play a little game: the teacher has prepared four chairs here, and five students are invited to come up. Who wants to go?
Teacher: Listen carefully to the requirements. The teacher said, after the start, please all five of you sit in the chairs, and everyone must sit down, okay? (good). At this time, the teacher faced the whole group and turned his back on five people.
Teacher: Let's begin.
Teacher: Are you all seated?
Health: Sit down.
Teacher: I didn't see them sitting, but I'm sure: "No matter how you sit, there are always at least two students sitting in a chair." Am I right?
Health: Yes!
Teacher: Do you want to know why the teacher made such an accurate judgment? In fact, there is an interesting mathematical principle-pigeon hole principle.
Second, experimental exploration.
Step 1: Study how to put four pencils into three pencil boxes. What are the different ways to put them? What interesting phenomena can you find from these methods?
1. (Show) Teacher: What are the different ways to put four pens in three pencil boxes? What interesting phenomena can you find from these releases?
2. Teacher: Next, ask the students to do the experiment in groups and fill in the release methods and findings on the record card.
Release method
Writing case 1
Pencil box 2
Pencil box 3
A few sticks at most.
A
B
C
D
Our findings
3. Group report and communication.
(4,0,0)、(3, 1,0)、(2, 1, 1)、(2,2,0)
Health: Anyway, there are always 1 pencil boxes with at least 2 pencils in them.
Teacher: What do you mean by "always"?
Health: There must be.
Teacher: What do you mean by "at least"?
Health: Not less than 2 branches, maybe 3 or 4 branches.
Summary: Put four pencils into three pencil boxes, and there will always be at least two pencils in one pencil box. (2 or more at most)
4. Teacher: Put four pens and rice in three pencil boxes. No matter how you put it, there are always at least two pencils in a pencil box. This is the conclusion we found through practical operation. Then, can we find a more direct way to reach this conclusion by only putting one situation, at least finding out the figures?
Health: We found that if you put 1 pencil in each pencil box, you can put up to 3 pencils. No matter which pencil box you put the remaining 1 pencils in, there will always be at least 2 pencils in a pencil box.
(Student operation demonstration)
Teacher: This division is actually how to divide it first.
Student: Average score.
Teacher: Why should we average the score first?
Student 1: If you want to find out that there must be at least two in a pencil box, first divide the remaining 1 roots equally. No matter which pencil box you put, there will be a saying that "there must be at least two in a pencil box".
Health 2: In this way, you can be sure that there will always be at least a few pens in a pencil box only once.
Try to put the pens in each pencil box and put them evenly as far as possible. How to express it by formula?
4÷3= 1…… 1 1+ 1=2
How about putting six pencils in five pencil boxes? (Demonstration with pencil operation) 6 ÷ 5 =1...11+1= 2.
What do you think of putting seven pencils in six pencil boxes? ……
99 pencil cases 100 pencils?
The teacher asked: What laws have been found?
Summary: There are more pencils than pencil cases 1. No matter how you put it, there are always at least two pencils in a pencil box. (Talking to each other at the same table)
Step 2: Study the phenomenon that the number of pencils is not 1 more than the number of pencil cases.
1, Teacher: So far, do you want to continue your research? What other issues are worthy of our further study? (Students ask questions independently: If it is not greater than 1, what is the pigeon hole principle and so on. )
2. Teacher: If the number of pencils is not more than the number of pencil cases 1, but more than 2 or 3, how many pencils will there always be in a pencil case?
(Show: Put five books in two drawers. How many books will there be in a drawer? )
Students think independently, communicate in groups and report.
Teacher: Many students don't have school tools. What methods were used?
Student: Average score. Divide the five books into two drawers, put two books in each drawer, and there is one left. No matter which drawer you put it in, there are always at least three books in a drawer. Health: 5 ÷ 2 = 2... 12+ 1 = 3.
(Show: How about three drawers and five books? How about five drawers and eight books? )
5÷3= 1……2 1+ 1=28÷5= 1……3 1+3=4
Teacher: Why is at least number not "quotient+remainder"? (Group discussion, report)
4. Can we find the rule of finding the minimum number by comparing the observation formulas?
Number of items ÷ number of drawers = quotient ... at least remainder = quotient+1.
5. Summarize the pigeon hole principle. What is the key to using the pigeon hole principle? (Find out the number of objects and drawers) and read relevant materials.
A ÷ n = b...c (c ≠ 0) Put an object into n drawers, and there are always at least (b+ 1) objects in one drawer.
Third, apply the principle.
Please have a try. (Oral answer, pointing out what is the number of objects and what is the number of drawers)
(1) Six pigeons fly back to five dovecotes, and at least two pigeons will fly into the same dovecote. Why?
(2) Keep 13 rabbits in five cages. How many rabbits should be kept in the same cage?
(3) Five bags of biscuits, each bag 10 yuan, for six children. How many cookies can a child always get?
2. Is the following statement correct? Tell me your reasons.
There are 370 sixth-grade students in Xiangdong Primary School, including 49 students in Class 6 (2).
At least two students in the sixth grade are in the same Amanome.
(370 items, 366 drawers)
Only five students in Class Six (2) have birthdays in the same month.
(49 objects, 12 drawers, "only" means certain)
C, 6 (2) At least 25 students are of the same sex.
3. Play the game of "guessing poker".
Draw five cards, at least a few of the same suit? 5÷4= 1…… 1 1+ 1=2
Painting 15. How many numbers are the same? 15÷ 13= 1……2 1+ 1=2
4. Students write down the phenomena that can be explained by the pigeon hole principle in life.
Careful observation+careful thinking = great discovery
Fourth, the whole class summarizes.
The fourth part of the teaching design of the open class of the sixth grade mathematics "Pigeon Cave Principle": P70-7 1 case 1, case 2, finish the problem-solving exercise 12/and 2.
Learning guidance goal
1. After going through the inquiry process of "pigeon hole principle" and having a preliminary understanding of "pigeon hole principle", we will use "pigeon hole principle" to solve simple practical problems.
2. Feel the charm of mathematics through the flexible application of pigeon hole principle.
Key points of study guidance: experience the inquiry process of "pigeon hole principle" and get a preliminary understanding of "pigeon hole principle".
Difficulties in learning guidance: understanding "pigeon hole principle" and "simulating" some simple practical problems.
Preview study plan
Have the students ever played poker? How many colors are there in playing cards? Take out two trump cards and draw five cards at random from the remaining 52 cards. I don't look at cards. I can say for sure that at least two of these five cards are of the same suit. Can you believe it?
Study Guide
What do you want to know through today's study?
Explore new knowledge of autonomous operation
(1) Activity 1
Courseware demonstration:
How many ways can I put three books in two drawers? Please put it aside and share your thoughts in the group.
1, students begin to operate, and teachers patrol to understand the situation.
2. Reporting communication and reasoning activities
What did you find? Who can talk about it?
Write the numbers on the blackboard according to the students' answers. Blackboard: (3,0) (2, 1) (1, 2) (0,3)
What other methods can be used to record? I use courseware to show what I recorded with pictures.
(1) Look at the records carefully again. What else did you find?
There are always at least two books in the drawer. )
(2) How can you draw a conclusion by putting it once? Inspire students to use the method of average score to lead to division calculation. ) blackboard writing: 3÷2= 1 (this) ... 1 (this)
(3) This method can quickly determine how many books are always in a drawer. (student exchange)
(4) Put four books in three drawers? Do you need a pendulum? Blackboard: 4÷3= 1 (Ben) ... 1 (Ben)
⑤ Courseware presentation: How about putting six books in five drawers?
Put seven books in six drawers?
9 drawers for 10 books?
Put 100 books in 99 drawers?
Blackboard: 7÷6= 1 (Ben) ... 1 (Ben)
10÷9= 1 (Ben) ... 1 (Ben)
100÷99= 1 (Ben) ... 1 (Ben)
6. What laws have you found by observing these formulas?
Students should say: at least number = quotient+remainder.
Teacher: Is this a rule? Let's have a try!
3. Deepen the exploration and draw a conclusion.
Courseware shows that seven pigeons fly back to five pigeon coops, and at least two pigeons will fly into the same pigeon coop. Why?
① Student activities
② Communication and reasoning activities
③ Is it "quotient plus remainder" or "quotient plus 1"? Whose conclusion is correct? Conduct group research and discussion.
Who can make it clear? Blackboard: 5÷3= 1 (only) ... 2 (only) at least = quotient+1.
(2) Activity 2
Courseware demonstration: put five books in two drawers. No matter how you put it, there are always at least a few books in a drawer.
Post-grouping report
Blackboard: 5÷2=2 (Ben) ... 1 (Ben)
7÷2=3 (Ben) ... 1 (Ben)
9÷2=4 (Ben) ... 1 (Ben)
So up to now, how do you think we can ensure that there are always at least a few books in a drawer?
(At least number = quotient+1)
I agree with your discussion. What we found is very interesting. Pigeon hole principle and pigeon hole principle are also called pigeon cage principle, which was first put forward by German mathematician Dirichlet in19th century, so it is also called Dirichlet principle. This principle is widely used in practical problems. Can solve many interesting problems. Let's have a try, shall we?
Flexible application to solve problems
1, explain the game questions raised before class.
2. Eight pigeons fly back to three dovecotes. No matter how you divide it, there are always at least a few pigeons in a dovecote.
3. In any 13 person, at least two people have the same birth month. Why?
Among any 367 students, there must be two students whose birthdays are the same day. Why?
Talk about feelings: Students, what do you feel about this class today?
Classroom detection
fill (up) a vacancy
1, 7 pigeons fly into 5 dovecote, and at least () pigeons will fly into the companion's dovecote.
2. There are 9 books. To put them in two drawers, there must be at least () books in one drawer.
3. There are 73 students in two classes in Grade Four, and at least () students in these two classes were born in the same month.
4. Give three different natural numbers at will, and the sum of the two numbers must be ().
Second, choose
1, five people spent 30 1 yuan on shopping, each spent an integer, and at least one of them spent not less than () yuan.
a、60 B、6 1 C、62 D、59
2. The total price of three commodities is 13 yuan, the price of each commodity is an integer, and the price of at least one commodity is not less than () yuan.
A, 3 B, 4 C, 5 D, uncertain
Third, solve the problem.
At present, there are five locks 1, 1, and all the keys can't be locked together. How many times can I at least try to match all the locks?
There are 5 boys and 5 girls in each group of Class 1, Class 6 and Class 4. Change their names to 10 respectively. How many digits can we call them two boys or two girls?
Development after class
1 class 6, class 2, 35 people. How many exercise books must Miss Li prepare at least to ensure that one person has more than two exercise books?
2. From 1, 2, 3... 100, among these 100 continuous natural numbers, randomly select 5 1 different numbers, two of which must be coprime. Why?
blackboard-writing design
drawer principle
5 ÷ 2 = 2 ... There are at least 3 1.
7 ÷ 2 = 3 ... There are at least 4 1.
9 ÷ 2 = 4 ... There are at least 5 1.
At least 611÷ 2 = 5 ...1.
At least number = quotient+1
The fifth teaching goal of the sixth grade mathematics "Pigeon Cage Principle" teaching design;
1. Make students understand some basic principles in extraction problems and solve simple problems.
2. Understand the relationship between mathematics and daily life, understand the value of mathematics, and enhance the consciousness of applying mathematics.
Teaching focus:
Pick up the question.
Teaching difficulties:
Understand the basic principle of extracting problems.
Teaching process:
First, create situations and review old knowledge.
1, show the review questions:
Teacher: Teacher, here is a question. I don't know which classmate can help me solve it.
2. Courseware demonstration: Put three apples in two drawers, and there are always at least two apples in one drawer. Why?
3. Students can answer freely.
Second, teaching examples 2
1. Display: There are four red balls and four blue balls of the same size in the box. If you want to touch the ball, you must have two balls of the same color. How many balls do you have to touch at least?
(1) Organize students to read the questions and understand the meaning of the questions.
Teacher: Can you guess the result?
Let the students guess and communicate with each other.
Name the students to report.
Students may answer when reporting: just touch four balls and touch at least five balls. ...
Teacher: Can it be verified?
The teacher took out the prepared red balls and blue balls, and organized the students to touch them on the platform to verify the correctness of the report results.
(2) Teacher: Just now we came to a conclusion through verification. How does this question relate to what we have learned before?
2. Organize students to discuss and communicate with each other. Then report to the students by name.
Teacher: The problem above is the drawer problem. Please recognize: what is a drawer? How many drawers are there?
Organize students to discuss and communicate with each other.
Call the students to report, and let them know how many colors the drawers are. (blackboard writing)
Teacher: Can you answer with the knowledge of the example 1
Organize students to discuss and communicate with each other.
Name the students to report.
Let the students understand that as long as there are more objects than drawers, there will always be at least two balls in the drawers. So make sure to draw two balls with the same color, and the number of balls drawn is at least one more than the number of colors.
(3) Organize students to discuss the problem-solving process, communicate with each other and understand the problem-solving methods.
It is not difficult for students to find that as long as they touch more balls than their colors 1, they can ensure that the two balls are the same color.
Step 3 do this
Question 1.
1, think independently and judge right or wrong.
2. Communicate with classmates and explain the reasons. Among them, "2 out of 370 students must have the same birthday" is the same as the "Pigeon Cave Principle" in Example 1, and "5 out of 49 students must have the same birth month" is the same as Example 2. Teachers should guide students to turn "birthday problem" into "drawer problem". Because there are at most 366 days in a year, if these 366 days are regarded as 366 drawers, and 370 students are put into 366 drawers, the number of people is greater than the number of drawers, then there are always at least two people in a drawer, that is, their birthdays are the same day. There are 12 months in a year. If this 12 month is regarded as 12 drawers, and 49 students are put into 12 drawers, 49 ÷ 12 = 4... 1 Therefore, there are always at least five in one drawer.
Third, consolidate the practice.
Complete exercise 12, questions 1 and 3.
Fourth, summarize and evaluate.
1, Teacher: What did you gain or feel from this class?
Verb (abbreviation for verb) assigns homework.
1. Do it. Mix 10 red, yellow and blue. If you close your eyes, how many sticks can you take out at least once to ensure that there must be two sticks of the same color? Are you sure there are two pairs of sticks of the same color?
2. Give it a try. Colour each box below red or blue. Look at each column. What did you find? What will happen to the conclusion if only two columns are drawn?
3. Expanding exercises (optional)
(1) Give five nonzero natural numbers at will. Some people say that we can find three numbers, so that the sum of these three numbers is a multiple of three. Do you believe it or not?
(2) Circle the eight numbers 1 ~ 8. On this circle, the sum of three adjacent numbers must be greater than 13. Do you know the mystery?