The first part of the teaching record of multiplication and division: the example on page 54 and related exercises on page 55 of the standard experimental teaching material for compulsory education in grade four (Volume II) published by Jiangsu Education Publishing House.
Teaching objectives:
1. Starting from students' existing experience, through observation, analogy, induction, verification and other activities, lead students to experience and explore the process of multiplication and division, and understand and master multiplication and division.
2. Deepen and enrich students' understanding of the law of multiplication and distribution through transformation and association, and enhance students' interest in learning mathematics.
3. Infiltration? From special to general, and then from general to special? Objective To understand things, cultivate students' awareness of finding problems and actively exploring, and improve students' mathematical thinking ability.
Teaching process:
First, collect materials by solving practical problems.
1. Solve practical problems in two ways and collect relevant formulas.
Teacher: Through the communication before class, the teacher has got a preliminary taste of the children in Class 4 (1 1). When the bell rings, the teacher believes that we will see more wonderful performances. Please look at such a question first.
Courseware demonstration:
Teacher: Read the questions softly. Please raise your hand if you can solve them.
Health: 70? 5+40? 5。 (blackboard formula)
Teacher: Can you elaborate on your statement?
Health: 70? Five is the money for five coats, 40? Five is the money for five pairs of trousers, which adds up to five coats and five pairs of trousers.
Teacher: The thinking is clear. (Many students raised their hands again. It seems that some students have different ideas. Let's listen together.
Health: (70+40)? 5。 (The teacher wrote it on the left side of the previous formula)
Teacher: What do you think of this formulation?
Health: Five coats and five pairs of trousers, which is five sets of clothes. I work out the money for buying a suit, that is, 70+40, and then multiply it by 5 to work out the money to pay.
Teacher: The students in our class worked really hard and soon came up with two methods. Then please look at the second question.
Courseware demonstration:
Health 1: 12? 30+ 16? 30。 I work out the kilograms sold in the morning, then the kilograms sold in the afternoon, and then add them up to get the kilograms sold every day. (The teacher corresponds to the formula in front of the right)
Teacher: Do you agree with him?
Health (anger): I agree!
Student 2: (12+ 16)? 30。 I calculate how many bags of rice I sell every day, and then multiply them by 30 to calculate how many kilograms of rice I sell every day. (The teacher corresponds to the formula blackboard on the left in front)
Teacher: Yes, with the experience just now, it is better now.
2. Observe the respective characteristics of the left and right sides of the two groups of formulas.
Teacher: Students, look at these formulas. The teacher found that the two formulas on the left felt quite similar. what do you think? (Students nod their heads in agreement) Then can you tell me where they are looking?
Sheng 1: The expression on the left has parentheses.
Health 2: multiply all the expressions on the left by a number outside the brackets.
Health 3: I can sum up what they said, that is, the formula on the left is to add two numbers first and then multiply them by one number.
Teacher: The speech is very standard. Let's look at the two formulas on the right. Do they have the same place?
Health 1: The product of two numbers is calculated first, and then added.
Health 2: I want to add that one of the two numbers is the same when multiplied.
Teacher: That's true!
3. Guide students to verify and form equations with left and right formulas.
Teacher: Students, the two corresponding formulas only use different ideas to solve the same problem, and the results should be equal. How can we know whether the results of two formulas are not equal?
Health: Calculation.
Teacher: Good method. (Teachers and students * * * the same mouth to calculate the first set of formulas)
Teacher: Through calculation, the left and right sides of the first set of formulas are equal to 550. In mathematics, we can relate them with an equal sign. (The teacher connects the first set of expressions with an equal sign.)
Teacher: Then let's look at the second set of formulas. Let's make a request. Who has the ability to make a judgment without accurate calculation? We can discuss with each other.
(Students discuss)
Health: 12 in the formula on the right? 30 is 12 30, 16? 30 is 16 30, which adds up to 28 30; The formula on the left is exactly 28 30, so it is equal.
Teacher: Very wonderful! Starting from the meaning of multiplication, the problem is also explained. Anyway, now we can safely write an equal sign between every two formulas. (The teacher connects the second set of formulas with an equal sign)
Second, explore the law and fully understand the connotation of the law of multiplication and distribution.
1. Observe the relationship between the left and right sides of the formula, guide students to observe the first set of formulas, and so on to the second set of formulas.
Teacher: Drawing the equal sign is not the end of our study, but the beginning of our research. The teacher is thinking that the results of these two formulas are equal. Is there any connection between the formulas? Let's read each equation softly again and see what we find.
(Students read the formula softly)
Health: In the first equation, the sum of 70 and 40 is multiplied by 5 on the left, and 70 and 40 are multiplied by 5 on the right, and then the two products are added.
Teacher: The crux of the problem is that after this change, the result of calculation is
Health (gas): equality.
Teacher: Yes, let's look at the second equation with this idea.
Health: The formula on the left is the sum of 12 and 16 multiplied by 30, and the formula on the right is 12 and 16 multiplied by 30 respectively, and then the results are the same.
Teacher: Students, is it a coincidence that the left formula of these two equations is first added and then multiplied, and the right formula is first multiplied and then added, which changes the operation order, but the result remains the same?
Health: No!
2. Teachers and students write a set of formulas with the same characteristics as the above formulas.
Teacher: Since everyone is so sure, now that the teacher has written a formula, can you write a formula equal to its number soon?
Blackboard: (15+ 10)? four
Health: 15? 4+ 10? 4。 (corresponding to the previous formula on the blackboard)
Teacher: Are the results equal?
Health 1: We can calculate separately. The formula on the left is equal to 100, and the formula on the right is equal to 100, so it is equal.
Student 2: Teacher, I want to talk about my own ideas. I can find that they are equal without calculation. The formula on the left represents 25 4s, and the formula on the right is 15 4s plus 10 4s, which is also 25 4s, just equal.
Teacher: Hey! Looks like you really found something. These are three equations with this rule?
Health: More than that.
Teacher: How many?
Health: Countless.
Teaching Record of Multiplication and Division People's Education Edition II [Teaching Content]
Primary school mathematics (Jiangsu Education Edition), the standard experimental textbook of compulsory education curriculum, Volume VIII, pages 54 ~ 55.
[Teaching objectives]
(1) experienced the process of exploring multiplication and division in specific problem situations, and felt, understood and summarized multiplication and division independently.
(2) To enable students to develop the ability of comparison, analysis, abstraction and generalization in the process of discovering laws, enhance their awareness of expressing mathematical laws with symbols, and further understand the relationship between mathematics and life.
(3) Infiltrate the method of understanding things from special to general, and then from general to special, so as to enhance students' interest and self-confidence.
[Teaching Emphasis and Difficulties]
Guide students to discover and understand multiplication and division.
[Teaching resources]
Calculator, multimedia courseware, physical projector.
[Teaching process]
First, create a situation and initially perceive the equation.
Teacher: Students, do you like to play games that break through obstacles? Want to play with the teacher?
The first level of the game: see who is fast.
Please look at the big screen:
(Show the picture and look at the questions to understand the information and questions: a short-sleeved shirt, 35 yuan, a pair of pants, 45 yuan, a coat, 65 yuan. Aunt Wang bought five jackets and five pants. How much will it cost? )
Teacher: Can you list the formulas quickly and get the results?
(Students calculate independently. When reporting, they asked different students to come up with different formulas, and told them the idea of solving the problem and the result of the formula. )
Teacher: Did the students use two different formulas to find out the answer? How much does it cost to buy a * *? , the result is 550 yuan. Then can you make these two formulas into an equation? Write it.
After the students finished writing the report, the teacher wrote down the equation on the blackboard.
Teacher: If the teacher changes the question to: Aunt Wang buys two short-sleeved shirts and two pairs of trousers, how much will it cost her to buy one? Will you solve the problem in two ways? Please work out the result quickly.
Student report.
Teacher: Can these two formulas be written as equations?
Health: Yes.
Teacher: Why?
Health: Because these two equations are all about how much a * * * has to pay, the result is the same.
Teacher: How to write it?
Health: (35+45)? 2=35? 2+45? 2。
Teacher: We passed this level successfully, and everyone got a score of 100. We successfully entered the second level to see who can see it correctly.
Thinking: In order to improve students' interest in learning, I started the teaching of this class by using games that break through customs and interspersed with life situations, which increased the interest of the class, stimulated students' learning motivation and laid a good cognitive and emotional foundation for students to further explore the law.
Second, observe and think, and feel contact independently.
Teacher: first observe the first equation: what is the relationship between the formulas on both sides of the equal sign?
Health: It's all three numbers: 45, 65 and 5.
Health 2: They got the same result.
S3: Do they all have them? Multiply by five. ?
Teacher: What do you multiply by 5?
Health: The sum of 45 and 65 on the left is multiplied by 5, and 45 and 65 on the right are divided.
Teacher: Do you understand? Who will tell me again, what is the connection between these two formulas?
Health: One is 5 times the sum of 45 and 65, and the other is 65 times 5 plus 45 times 5. They got the same result.
Teacher: I am really observant, so what is the connection of the second equation?
Health: In the second equation, the formula on the left is 5 times the sum of 35 and 45, and on the right is 35 and 45 divided by 5 and then added, so the left and right numbers are the same.
Teacher: The students did well, and everyone got 120. This discovery enabled us to successfully enter the third level: to see who wrote it correctly.
Thinking: Students' learning is a process of self-construction. In this process, students gradually change from concrete and individual appearances to abstractions. Without the perception of individual characteristics, there is no overall perception of a class of things. This link enables students to observe that the left and right sides of each formula have the sum of two numbers multiplied by a number, which is equal to the result of two numbers multiplied by this number and added separately. It is difficult to express the relationship between formulas in language in this lesson, and we can't get away from the point in order to break through the difficulties. Therefore, in order to let students freely express their findings, teachers should also do some integration and optimization in a timely and appropriate manner, but not too demanding, and may not be able to draw conclusions from books. The rigor and standardization of language can be solved step by step in the next class. By choosing something, doing something, not doing something, or doing something later, this is the natural way to teach a lesson.
Third, create equations and deeply understand the law.
Teacher: Can you write several sets of such equations?
Students write equations independently, and teachers patrol for guidance.
Teacher: Students in the same position should check with each other to see if their positions are correct and the formulas in each group are equal.
Students report after group activities.
Teacher: What's the result of the formula you just wrote?
Health: The result is equal.
Teacher: Are there many such formulas?
Health: A lot.
Teacher: Good. Everyone will get 150. Now let's go to the fourth level and see who can make it clear.
Thinking: Learning is a complicated psychological internalization process. Being able to tell the characteristics of both sides of these two equations is not necessarily internalized into your own understanding, but can also be tested by writing, that is, creating an equation with the same form as the observed equation, and further internalized. This is the process of putting forward hypothesis, and it is also the first time to establish the representation of multiplication and division law in students' minds. Whether this law is universal or not, we need to find out this formula in form now, and provide material for the next verification.
Fourth, summarize and express the law abstractly.
Teacher: Look at these equations. Have you found anything?
The students were lost in thought.
Teacher: What are the characteristics of these equations on the left and right? What was the result? Please discuss in groups and share your findings with others.
Group communication.
Teacher: Which team reports first?
Health: We found that the formula on the left side of the equal sign is the sum of two numbers multiplied by the third number, and on the right side, the two numbers are multiplied by the third number separately, and the products are added again. The results of these two formulas are equal.
Teacher: Do you understand? Who will talk about your group's findings again?
Student: We found that the sum of two numbers multiplied by one number equals the sum of these two numbers multiplied by that number respectively.
Teacher: Do students in other groups feel the same way?
Health: Yes.
Teacher: Then can you express your findings in other ways?
Health: (a+b)? c=a？ c+b？ c
Teacher: What do the letters here stand for?
Health: A and B represent two numbers that are added first, and C represents the number that both have to be multiplied by it.
Teacher: You mean that A and B stand for two addends and C stands for a multiplier, right? How else can I express it?
Health: (+□)? ○=? ○+□? zero
Teacher: The laws we found can be expressed by letters or by graphs. This law is the law of multiplication and distribution. Write on the blackboard. This is the most difficult hurdle to break through. We successfully passed the obstacle. How many points do you want to give yourself?
Health: 180.
Teacher: The teacher thinks it should be 800 points.
Hallows: Yes!
Teacher: But if you learn it, you can use it. Let's enter the fifth level: see who uses it well.
Thinking: The discovery of a rule needs to extend from individual to class, and then verify and affirm it. Because the multiplication and division method is abstract, the requirements for normative language in the new textbook are lower than those in the old textbook, and it is difficult for students to describe it completely in the standardized language in the book, so I also asked questions according to the textbook in my design: What did you find? The expression of this discovery is personalized and can be expressed by symbols or language. As long as students find and understand the law, of course, the teacher's summary should be standardized, and the teacher should pay attention to guiding students to express multiplication and division in a standardized way in future teaching.