? Division comes from "average score". In the division in the table, the children learned two meanings of division-"equal division" and "inclusive equal division". In daily life, when things are evenly divided, the results include two situations: one is the situation that they are just divided and there is no remainder (that is, the remainder is 0), which is what division in the table involves; One is that there is a remainder after the average score (the remainder is not 0), which is the content of remainder division to be studied.

This content mainly has two parts: the first part is the meaning and calculation of division with remainder; The second part is to solve the problem. There are two problems to be solved in the meaning of remainder division, one is what is remainder division, and the other is the relationship between remainder and divisor, which correspond to example 1 and example 2 in the textbook respectively.

Example 1 By dividing 6 strawberries and 7 strawberries on average, let children realize the two situations of dividing things, and then establish the concept of remainder division. However, the teaching demonstration shows that the example of 1 will cause some interference to children's future teaching. With the help of textbook example 1, the reasons for interference are analyzed as follows:

? The purpose of the example 1 is to let children understand that in daily life, when things are equally divided, the results include two situations: one is the situation that has just been divided, and there is no surplus at this time; One is that there is a surplus after the average score, and then it leads to what we are going to learn in this class-division with a remainder.

? One plate for every two strawberries, six strawberries, and three plates. In other words, three plates are needed. This is completely understandable to children.

? Put a plate for every 2 strawberries and 7 strawberries. It was found that there was 1 strawberry left after three sets. The textbook specifically reminds us that the remaining 1 is a remainder. In other words, three plates are needed, and the other strawberry has no plate and is placed outside the plate. For this, children can fully understand.

? So, where is the interference? Please look at the textbook example 5:

? Twenty-two students go boating, and each boat can take up to four people. How many boats do they need to rent at least?

22÷4=5 (ship) ... 2 (person). Then it is analyzed that the remaining two people also need to rent a boat, so at least 5+ 1=6 boats should be rented. At this point, children can still understand it without any difficulty.

Now, let's combine Example 1 with Example 5. Suppose we ask-how many plates do we need to hold these strawberries? At this time, the child will answer that 3+ 1=4 plates are needed. Then, will the child ask such a question: Teacher, do you want to put the remaining 1 strawberry on a plate? Why do you want it in the future and don't want it in the future? This is the interference element ...

? Let's take a look at the arrangement of Example 2 in the textbook:

? Using sticks to build an independent square, we can find that with eight sticks, just two squares can be built. In other words, 8 branches are divided into 2 parts, each with 4 branches, and there is no surplus; Nine branches, 10 branches, 1 1 branches, and there is a surplus after the average score. It can be seen that Example 2 can completely assume the function of Example 1, which leads to the content of "Division with Remainder". At the same time, example 2 also carries its own function-by putting an independent square with a stick (the remainder is less than the divisor), let the children understand the relationship between the remainder and the divisor.

Based on the above analysis, I deleted the teaching content of Example 1 in this class, and all the teaching was carried out with Example 2.

? In addition, division with remainder also involves periodicity. In other words, the problem of "periodicity" is an important aspect that division with remainder must face and study. Textbook Example 6 deals with the study of periodicity:

In the "periodicity" question, you can use the remainder to determine the relevant position question and answer what color the flag is.

? Based on the need of learning periodicity, I designed the pre-class conversation part. Let the child count with his left hand, and decide which finger the number falls on when counting. At the same time, it can stimulate children's interest in learning. Let the classroom dialogue better serve the next class. Stimulating children's interest in learning is not just a simple active atmosphere. How to better serve the next teaching is also a problem that I have been thinking and studying in recent years.

Instructional design:

Teaching content People's Education Edition, Mathematics Volume II P59-60 cases, 1, 2, Do something and exercise 14, 1, 2.

Teaching material analysis's content is the extension and expansion of the knowledge of division within the table, which is taught on the basis of division within the table. The textbook pays attention to connecting students' existing knowledge and experience, and selects a few familiar things as examples in combination with specific situations, and is accompanied by physical charts to let students understand the significance of remainder division.

The analysis of learning situation and the understanding of division with remainder are based on the fact that students have learned multiplication and division in tables. Students have just learned the division in the table in the previous stage and have been exposed to many completely completed examples. The thinking of sophomores is mainly figurative thinking. To complete the transformation from thinking in images to thinking in abstract logic, it is necessary for students to experiment and experience the formation process of knowledge through hands-on operation. In teaching, we should acquire knowledge and develop students' abstract thinking by accumulating observation, operation, discussion, cooperation and exchange and abstract generalization according to the systematicness of knowledge and the thinking characteristics of junior two students.

Teaching objectives

Knowledge and skills: make students experience the process of abstracting the remaining phenomenon after average score into division with remainder, initially understand the meaning of division with remainder, and know the remainder.

Mathematical thinking: Through operation, observation, comparison and other activities, students can find that there is a remainder in the division of things in life, so as to understand the significance of remainder and division with remainder, and initially cultivate students' comprehensive thinking consciousness.

Problem solving: Know the division with remainder, strengthen the concept and master the algorithm. Can write the division formula according to the average residual activity, and correctly express the quotient and remainder.

Emotional attitude: Infiltrate the consciousness and methods of intuitive research, cultivate students' ability of observation, analysis and comparison, and let students feel the close connection between mathematics and life.

The focus of teaching is to abstract the remaining situation after average score into division with remainder.

Difficulties in teaching: Understanding the significance of division with remainder.

Teaching preparation courseware and stick

teaching process

First, talk before class:

People have two treasures, hands and brains. Hands can work and brains can think.

You can't do things well by replacing your brain with your hands. Without your hands, you can't do anything well with your brain.

Use your hands and brain to create. ...

PPT shows the picture on the left, counting from the thumb to the little finger in the order of natural numbers, and then circulating the numbers in turn, marking the data to 32. ...

Teachers have a magical ability. As long as you quote a number, I will know which finger it is on. Can you believe it?

Through this game, children's enthusiasm for learning is mobilized, and the problem of "periodic" division research is also infiltrated. And it corresponds to the verification link of putting a square with a small stick in the new lesson teaching. (In other words, in the process of asking students to verify whether the stick is correct, we should pay attention to the topic of conversation before class so as to blend in with each other ...)

Second, explore new knowledge:

Preliminary feeling, teaching example 2:

1. Recall the meaning of division in the table (just after the average division). The child moved and the teacher drew a picture on the blackboard.

How many squares can you make with eight sticks? Can you use a formula to express the process of pendulum just now?

8-4-4=0,? 8÷4=2

Two communication: one is to communicate the relationship between subtraction and division, and realize that the division formula in mathematics comes from the subtraction formula, and the subtraction formula is a simple record of the subtraction formula. The second is to communicate the relationship between graphs and formulas (subtraction formula and division formula). Let the children understand the relationship between the parts and establish a dynamic picture of feelings in their minds.

2. Understand the meaning of division with remainder (the case that it cannot be divisible). It was the child's action, and the teacher drew a picture on the blackboard. Hands-on operation feels that the average score will be surplus.

How many squares can you put in a square with nine sticks? Notice, what's the difference between these eight sticks? Can you use a formula to express the process of pendulum just now?

(1) (think and discuss in groups) Show students' representations and compare them.

9-4-4= 1 (root). 9÷4=2 (square) ... 1 (stick)

Tell me what this formula means. Communicate the relationship between formula and graph again. The communication between teachers and students is divided into two levels. 1 from number to number, 2 from number to number. Then, under the guidance of the group leader, the children in each group pointed to each other and said ...

Summary: This formula indicates that if you make a square with 9 sticks, you can make two squares, and there are more than/kloc-0 sticks. The ellipsis indicates the remainder, and 1 is an unnecessary number, which we call the remainder. What does the remainder mean? (indicating the insufficient part)

(2) Comparison and induction to improve the cognitive structure.

Observing and comparing two formulas, 8÷4=2 (pieces) and 9÷4=2 (pieces) ... 1 (root), lead students to realize once again that there will be two situations when calculating division: one is that there is no surplus after all, and the other is that there is surplus after division, but it is not enough except the rest.

(3) Constantly throwing sticks, requiring constant upgrading. How many squares can be made with 1 1 sticks and 13 sticks respectively? How many sticks are left?

1 Draw charts and write formulas in a notebook to communicate the relationship between charts and formulas.

Imagine drawing a picture in your mind, then drawing the picture in your mind in a notebook, writing a formula and communicating the relationship between the picture and the formula. The group leader leads the children to talk to each other. )

(4) Continue to insert a stick to understand the relationship between remainder and divisor.

How many squares can you put with 17 and 18? How many sticks are left?

Discussion, can you verify that your conclusion is correct? How to verify? Teachers' classroom presupposition:

Teacher: How can you prove that you are right?

Health: 4× 4 =1616+1=17.

Let the students watch the teacher put a square with a stick, and then exchange the relationship between the picture and the formula again.

Teacher: According to the result of 17, we can infer the results of 18, 19 and 20.

The students answer together, and the teacher writes on the blackboard.

18 ÷ 4 = 4 ... 219 ÷ 4 = 4 ... 3 20 ÷ 4 = (Dig a hole for the child and see if you will be fooled. )

Teacher: Why not 4...4? Where did the fundamental change take place? Quantitative change leads to qualitative change.

Teacher: One quantity is changing. Change another quantity. The original dividend is changing, which will cause the change of the remainder. The remainder can always be increased?

Health: The remainder is less than the divisor.

Teacher: On the other hand.

Student: The divisor is greater than the remainder.

Teacher: The divisor is 5. What is the remainder?

Health: 4,3,2, 1

Teacher: The divisor is 8. What is the remainder?

Health: 1, 2, 3, 4, 5, 6, 7

Teacher: □□□□ = □…………… The divider is A, the divisor is B, the quotient is C, the remainder is D, and the board A÷B=C……D(B≠0)D=4, B=?

Health: 5, 6 ...

Teacher: As long as one condition is met,: D is less than B.

Note: there is a jam in this teaching link, and the child's reverse thinking is a bit uncomfortable.

Fourth, go back to the game itself and reveal the mystery of the game. What is the remainder? This is on the first finger. When the data is relatively large, the calculation results can be determined by column vertical method, and then the teaching content of the next lesson-vertical division is introduced.

Note: the same topic is-circulation on the calendar, drinking quiz, textbook example 6.

Verb (abbreviation of verb) consolidation exercise: P60 "do":

1, students write their own books, and fill them out 1 question. Feedback communication: 17÷2=8 (group) ... 1 (individual) 23÷3=7 (group) ... 2 (individual) Tell me what the quotient and remainder of these two formulas are and what they mean respectively.

2. Complete the second question. Use the learning tool to put a pendulum as required, and then fill in the blanks according to the results of the pendulum. Show the fill-in-the-blank situation of individual students, talk about what the quotient and remainder in each question represent respectively, and emphasize the unit names of quotient and remainder.

Understanding of division and remainder in blackboard writing design

8÷4=2 (piece)

9÷4=2 (pieces) ... 1 (root)

1 1÷4=3 (pieces) ... 3 (root)

17÷4=4 (pieces) ... 1 (root)

18÷4=4 (pieces) ... 2 (root)

19÷4=4 (pieces) ... 3 (root)

20÷4=5 (pieces)

The remainder is less than the divisor, and the divisor is greater than the remainder.

Reflection after class

First of all, in order to break through the teaching difficulties of this course, I mainly took the following three measures:

1. Promote students' understanding of new knowledge through intuitive operation. In teaching, with the help of intuitive operation, from intuitive operation to symbolic representation, the understanding of the concept of remainder and the meaning of division by remainder enables students to understand what they have learned from many aspects and angles, establish the relationship between operation process, language expression and symbolic representation, and realize students' real understanding of mathematical concepts.

2. Help students understand the meaning of division with remainder through comparison. The first is the comparison of the process of dividing things equally. Through the operation process of "eight sticks and nine sticks make a square", students can feel that there are two situations in the process of dividing things equally, that is, there are two situations after dividing things equally. In comparison, students can expand their understanding of division and better understand the meaning of remainder and division with remainder. Secondly, the horizontal forms of division with remainder and table division are compared. By combining the operation process, let the students understand the names of each part and the meanings of each number in the division horizontal with remainder. Through this comparison, students can not only arouse their existing knowledge and experience, deepen their understanding of division with remainder, but also feel the connection between knowledge, provide support for building a reasonable knowledge structure network, and cultivate their ability of analysis, comparison and induction.

3. Combined with related examples and exercises, provide students with opportunities as much as possible to experience the process of discovering and abstracting mathematical problems from real life or specific situations, so as to accumulate the experience of discovering and asking questions for students, cultivate students' awareness of problems and sensitivity to mathematical problems, embody the basic idea that mathematics comes from life and then returns to life, and strengthen the close relationship between mathematics and life. There are still some shortcomings in the actual teaching process of this course. For example, students can't do oral calculations quickly in the later practice, so they should do more exercises in this area before class; After students begin to operate, there is no chance for them to fully communicate and express themselves. Therefore, in the future teaching, students should be allowed to describe their ideas and the process of hands-on operation in their own language, so as to effectively improve their hands-on operation level and thinking expression ability.

Second, there are the following problems in this course, which need to be improved:

1. When teaching 17 stick and 18 stick to form a square, children are too eager to write the division formula, resulting in some children not being able to write the division formula. We should still guide the children to draw pictures first and get the correct answer with pictures.

2. The remainder is less than the divisor, and this link goes too fast. In other words, it smells like a direct notification. It also leads to a little stuck in the later exercises. There should be multiple groups of division formulas for children to find rules. When the divisor is 4, the remainder is 1, 2,3 cycles, and then let the children understand that the remainder is less than the divisor.

3. The whole class is square, and the divisor is always 4, which is easy for children to form a mindset. As a result, some children can't walk out of the square. If the group is divided into triangles, squares and regular pentagons, can it improve the classroom to some extent?

4. □□□□ =□………………, this topic will be divided first, and the rest will be discussed; When the child has a certain clue, there will be a remainder and discuss the divisor. In other words, we should think positively before thinking reversely. In addition, □□ = □………………………………………………………………………………………………………………………………………………………………………………………………………………. Is the child counting the edges of the box and adding something? (I don't know, ask the children tomorrow ...)

5. In order for most children to master division with remainder, they must also have a special class to circle and draw a picture, and realize how much the quotient is, how much the remainder is, and what is the connection with dividend and divisor. Then you can transition to the skill training of calculating division with remainder by formula.

? Finally, thank you for your multi-angle and multi-thinking exchanges, discussions and comments after class. Teaching is an art with "regret", and it is precisely because of regret that it shows the infinite charm of the classroom.

It is because of regret that there is such a sentence-"Three thousand drowned, each take a gourd ladle." Every teacher can and should get his own "that ladle of water" in the "three thousand drowning"