Class record of division with remainder

The division with remainder in the preface is located in the second half of the second chapter of "Divider is the division of a number", which is a perfect understanding model of vertical division operation. This course is based on the dialogue between teachers and students, with students as the center and teachers as the guide and host.

Prepare before class. He Laoshi will issue a pre-test sheet two days in advance. Because I was attending a class in Long Mei, Zhengzhou, I received the prediction test paper sent by students before class, and found that students did not construct the meaning of vertical division very well. Five minutes before class, the class list was sent out, and the students began their math journey with great interest.

Review before the first class

Because it is a borrowing class, He Laoshi first made two agreements with the students in the "Jade Bird" class. One is to clap your hands and put everything down and sit up straight at once; Second, when a classmate answers questions particularly well, everyone applauds him.

After the appointment, He Laoshi and his classmates checked the review answers before class. The students are very enthusiastic. In the process of answering questions, a very interesting name-accounting suddenly appeared. He Laoshi quickly accepted this interesting name and used it to carry out the course more smoothly.

Classroom examples in the second section

First of all, the mathematical discussion.

Read the questions and requirements together. After reading the questions, the students immediately began to write down their ideas on the class schedule. He Laoshi toured below and found that many students had met with difficulties. )

He Laoshi, let a classmate explain it to you. The students write and explain on the blackboard.

Health: There are 256 photos, and 6 photos can be inserted into each page. Translated, six photos can be inserted into a page, so the division by six is greater than two, so we borrowed five and quotient four to write in the tenth place, and drew a horizontal line after four, six and twenty-four. There is still 1 left in the tenth place, which is not enough, so we pulled down 6 to become 16.

Then I wrote the horizontal body. "256 present 6 = 42 ... 4 pages.

Teacher: There seems to be some problem here. Let's put it down and make it clear and simple. Give him a hand first.

Other students responded with warm applause very sincerely. )

Health: Just now he said that it is closest to 16, and 368 is also closest to 16.

Health: 18 is bigger than 16, so you can't score.

Teacher: I think there are two problems with what he just said. One is that two places are not enough for 100 places and 10 places for business. How to borrow ten places?

Health: No matter how many businessmen are, they are all over 200, and they can't do hundreds of businesses.

Teacher: Why did you start business at the age of ten?

Health: Because you go to ten places to do business, you can get in from ten to one hundred.

Health: there are thousands in a hundred, so you can't divide them. If there are ten, you can leave a ten when it is full.

Teacher: I'm at a loss. Let's think it over. We judge that he is definitely not enough to divide by 100. 2 plus 5 equals 25?

Health: 25 ten.

Teacher: Good. Where can I get 25 10 divided by 6?

Health: Four, six and twenty-four, from ten businessmen.

Teacher: 24?

Health: 24 ten.

Teacher: Is it all over except ten o'clock? There are 25 tens left MINUS 24 tens?

Health: 1 10.

Teacher: Is the remaining 1 10 smaller than the divisor?

Health: small.

Health: Pull down 6 to become 16 one.

Teacher: Then what do you do with 16?

Health: Divided by 6.

Teacher: How much is the quotient?

Health: 2.

Teacher: Just now someone said 262,368. It seems that 18 is closer to 16? But what seems to be wrong with 18?

Health: 18 is greater than 16.

Teacher: So you can only do 2 quotient. 262.

Teacher: We use a horizontal line to indicate that this step is over, and then what?

Health: Yu 4.

Teacher: What does this 4 mean?

The students expressed their opinions noisily, and He Laoshi motioned the students to raise their hands to answer questions.

Health: There are four left.

Teacher: Why can I stay? Can you put four pieces of paper on one page?

Health: No.

Teacher: How many pages can we squeeze in?

Health: Six.

Teacher: Let's continue our discussion. What does 42 mean for the problem that just happened to that classmate? How to bring the unit?

Teacher: 42 means that 42 pages have been inserted, and 4 means that there are 4 pages left after 42 pages have been inserted.

After intense discussion in this part, students have basically understood and mastered the algorithm of vertical calculation.

Second, the algorithm summary.

He Laoshi turned the fill-in-the-blank question into a question, and the students scrambled to be the first to give the answer. In this way, the calculation rule of dividing three digits by one digit is summarized, and the overall acceptance of students is still very high.

Thirdly, the discussion about the relationship between the parts of division and the remainder.

He Laoshi and the students discussed the meaning of each number in the vertical form in the topic, and found that the students were somewhat unclear about the concepts of number of copies and number of copies.

He Laoshi asked students to read the topic "You can insert 6 photos per page, which can fill 42 pages." Try to make students understand the meaning of the question again to identify the number of copies and each copy.

Teacher: Every copy is every copy, right? What's that number?

Student: Divide it equally.

Teacher: So, how many pages did you insert?

Health: Six.

Teacher: How many pages did you insert?

Health: 42 pages.

Teacher: So how much is it per share?

Health: 42.

Teacher: Is it really 42 per share?

Teacher: Is it 6 copies or each copy?

Health: 6 per share. (Several students mutter to themselves)

Teacher: It is clearly stated in the title that six photos can be inserted into each page, that is to say, one photo can be inserted into every few photos?

Health: 6.

Teacher: six for one, and you said that six is not one. Is it or not?

Health: Yes. (Super loud and confident)

After this part of the discussion, students have a deeper understanding of the number of copies and each copy.

He Laoshi decomposed the topic into vivid animations, and discussed the relationship between the parts of division according to the meaning of each part in the topic.

Teacher: Can you see the relationship between the number of copies, each copy and the number of missing copies in the picture and the total?

As soon as the problem came out, only a few students knew it, and other students were a little puzzled, so He Laoshi found a group to discuss it, and toured below to communicate with his classmates from time to time.

Student: number of copies × number of copies+number of parts less than one = total.

Health: 42 copies, 6 copies each, 4 copies less than 1 copy, making a total of 256 copies.

As he spoke, He Laoshi helped to circle PPT. After explaining and drawing, the following students basically understood. The divisor × quotient+remainder = dividend is obtained by smoothing.

Teacher: This classmate speaks very well! Now let's check this vertical form according to this quantitative relationship and write it on the right side of the example.

While talking about He Laoshi, the students talked about the inspection process together and recorded it on the blackboard. He Laoshi also counted the correct rate of students. Most of the students did it right. Raise your hand. )

Teacher: Just now, a classmate found that multiplication and division seem to have something to do with it. What does it matter?

Health: It's the other way around.

Teacher: Let's have a look. What do the two multipliers here correspond to?

Health: quotient and divisor, product is dividend.

Teacher: What do you get when you add two numbers together? (talking and drawing)

Health: And. ?

Teacher: What is the process of multiplication and division? (gesture reminder)

Health: I pushed it back.

Fourth, unit discussion.

Teacher: There is one more question that we need to think about. What is the difference between these units? See who has strong thinking ability and can find problems that others can't.

Student: There are copies and numbers.

Student: quantity ÷ quantity = number of copies ... number.

Teacher: I said a little. I also took this course in our Chenguangshan school. A classmate said so. Teacher, I found that these three "ones" all have the same meaning, one, one each, one each. He said so.

Teacher: The teacher asked another question that has nothing to do with the example. If 256 photos are divided into six on average, what about each photo?

Health: 42.

Teacher: What's the 42 here?

Student: Numbers.

Teacher: What about 6?

Student: Number of copies.

Teacher: There are four left in the end. That's the number. That's all for today's class. Let's hand in the exercises at the back of the list. (Melodious bell rings, just finished the course)

After the postscript class, we had a big teaching and research.

Teacher Gan: First of all, the mystery when new problems appear. When the question first appeared, one of them was "How many cards are left?" Should not appear, why? Let's think about it.

Miss Li: Let the students think that it is still the original topic.

Miss Gao: Just give me a hint.

Teacher Gan: Yes, we have new questions, all of which are mysterious. Although this is a small problem, it will make students encounter problems in the future and must confuse them. Then, you must first use the existing model to analyze the problem. What is the existing model?

Teacher Gan: In multiplication, the existing model is: number of units × number of copies = total number; In division, it is: total number ÷ number of units = number of copies/total number of copies = number of units, then your model becomes a known total number and number of copies, what is each unit?

Miss Zhao: So it won't be bypassed.

Mr. Gan: Students should do it with the existing model, not teachers. Next, let the students calculate in the form of columns, and each student will talk about the principle. The total number is 256, divided into six points on average, and then the rules are made while walking. After that, there is a 4 left. What is this? The monster is out! We cut it down? It becomes a question of how to deal with this 4, and the concept of remainder comes out. This is called formula merging. You see, we are consolidating the previous formula and using it, a new model appears. If you add new tricks, it's not clear. If it is less than each copy, it is called remainder. The new rules of vertical division came out. And then me. Am I right? How to check? Checking calculation is also called inverse operation, but inverse operation is more difficult. Why? There is a remainder, so we must first change the understanding mode. This mode of understanding is the source of the rest. From the diagram, we can see that the number of copies × each copy+remainder = total, right?

Miss Zhao: It has enriched the previous model and changed it.

Teacher Gan: What is it if it is expressed by division? In fact, there are two forms, one is modeling. All the variants are inverse operations, reciprocal addition, subtraction, reciprocal multiplication and division, reaching the reciprocal understanding of the model, and then all the relationships are completed. In fact, the lessons after this lesson include extracurricular exercises and consolidation exercises, but this lesson has two aspects: the gestalt of understanding mode-the number of copies, the number of copies and the remainder expressed by division and multiplication must be related to the name of the division part, and the two should appear together. But why do we have a total? This total number is a model. Unify multiplication and division, completely unify all the names, and let the students fully understand and connect, and the whole meaning will be straightened out.

Miss Zhao: The total amount used is his gestalt. Combining the previous two into one, we are talking about the number of copies and each copy.

Teacher Gan: When you talk about the unit, if the students are unfamiliar with this course, it is actually very simple. You are wasting your time. See which units are the same. The total, remainder and units of each copy must be the same, but the units of each copy are different. Multiplication and division always have different units, and the units of addition and subtraction are the same. This is auxiliary, not the most important, it makes this model more clear and consistent. When instructing students, students can't get out. How to guide teachers at this time is actually an experience problem. If it is a foreshadowing problem, what are the similarities and differences? The problem was solved at once. What did you find?

Miss Zhao: You write the formula there, you observe it, ask it, and it comes out.

Miss Li: Sometimes people will understand that sentence by changing its expression.

Miss Zhao: Actually, if you list what Mr. Gan just said as a question, this class will be interesting. What the algorithm in this class summarizes is dead. So you described the problem-solving process, presented vertically. The presentation process is the model of the relationship among the number of copies, the number of copies and the total number of copies mentioned by Mr. Gan just now. Then ask, how to prove the rationality of your answer, and change those questions, and the whole class will be different.

After teacher michel platini's combing, I have a clearer understanding of the modeling of this course. First of all, according to students' cognition, we should make clear the goal of this class, that is, focus on difficulties and breakthroughs. The method is to start with students' existing knowledge, let them encounter problems, focus on problems, and finally let them form a new model. Another one has broken through my previous prejudice. I think only excellent and thoughtful students can give challenges. Traditionally, poor students can't be challenged because they will shrink back, but this is not the case. "Poor students" should have challenges, but not the kind that they can't do, but they can work hard, so that they can have a sense of accomplishment and become interested in mathematics.