1. Introduction. Present: There are 9 trees lined up on the side of the school playground, in order to beautify the campus environment, the students put a pot of flowers between each adjacent two trees, and there are no flowers at the head and tail, how many pots of flowers are placed in total?Students try to solve it, and the whole class communicates.
Student 1: I used the drawing method, and there were 9 trees in total, and there were 8 "empty", so I put 8 pots of flowers.
Teacher: This "emptiness" is mathematically called "interval". It is easy to see that there are 8 intervals between 9 trees by drawing a diagram, and if you know the number of intervals, you will know the number of pots of flowers.
Teacher: If there are 1,000 trees lined up, and there is still a pot of flowers between each two adjacent trees, and there are no flowers at the head and tail, how many pots of flowers are placed in total?
Students think independently and communicate as a class.
Sheng2: 1000 trees are lined up, and there are 999 intervals, so 999 potted flowers can be placed.
Teacher: How do you know that there are 999 intervals?
Birth 3: 9 trees have 8 intervals, so 1000 trees have 999 intervals. Teacher: This is a reasonable assumption, is there any other way?
Sheng4: You see, starting from the beginning, one tree and one pot of flowers, one tree and one pot of flowers, and finally the tree is very lonely, and there are no flower pots in the back, so the number of flower pots is 1 less than the number of trees in the tree, and a total of 999 pots of flowers can be placed.
Teacher: Do you understand what he meant?Sheng: I see.
Teacher: Even though the number has become larger, we can still use the method of drawing pictures to analyze problems. You can think about the problem like Sheng3: start from scratch, a tree corresponds to a pot of flowers, a tree corresponds to a pot of flowers, and finally the tree is very lonely, there are no pots to correspond to it, so the number of pots is 1 less than the number of trees in the tree, and a total of 999 potted flowers can be placed. Is this method good?
Sheng: Okay. Teacher: Mathematically, this method is called "one-to-one correspondence." With the help of drawing and "one-to-one correspondence", it is easy to find the relationship between the number of trees and the number of pots. Board book: one-to-one corresponding to the expansion.
1.Apply the idea of "one-to-one correspondence" to solve problems.
1) Teacher: If there are still 1,000 trees, and there is a pot of flowers between each of the two adjacent trees, and flowers are placed at the head and tail, how many pots of flowers can be placed in total?
Students think independently, and teachers and students communicate. Teacher: How many pots of flowers have been placed?
Raw (Qi): 1001 pots.
Teacher: Tell me what you think?
Sheng1: Just now, when "the head and tail do not put flowers", you can put 999 pots, and now there are 2 more pots of flowers at the head and tail, with 999 + 2 = 1001, so you put 1001 potted flowers.
Teacher: He linked the results of the above question and compared the differences in the placement methods of the two flower pots, and came up with 1001 pots, which is a very good way. Any other ideas?
Sheng2: I think like this, the beginning is a flower pot, and the end is also a flower pot, a flower pot corresponds to a tree, and so on, and finally there is a pot of flowers left, and the flower pot is 1 more than the tree, so 1000 + 1 = 1001.
With the help of the diagram, the method of "one-to-one correspondence" is used to illustrate: the number of intervals is 1 more than the number of trees
2) Teacher: If these 1,000 trees have flowers at the beginning and no flowers at the end, how many flowers will be put in total?Students think independently, and teachers and students communicate.
Hair flowers for this tree, a lifetime fan corresponds to a tree, so a trace of rejection and down, so the number of pots is as much as the number of trees, put 1000 pots of flowers.
With the help of the diagram, the "one-to-one correspondence" method is used to illustrate that the number of intervals is as large as the number of trees.
3) Teacher: If there are 51 trees in a row, and 4 pots of flowers are placed between each of the two adjacent trees, with no flowers at the head and tail, how many pots of flowers should be prepared?
Students think independently and try to solve problems.
Student board performance: (51-1) x4 = 200 (pot).
Teacher: What is the "51-1" here asking for?
Birth 1: The number of trees.
B2: No, it's not the bare number of the tree, it's the interval number.
Teacher: Yes, the question has already told us that the number of trees is 51, so why do we use "51-1" to find the interval?
Sheng2: Because "the head and tail do not put flowers", the beginning is a tree, and the end is also a tree, there are a total of 51 trees, there are 50 intervals, so you have to use 51-1=50.
Then with the help of the diagram, the method of "one-to-one correspondence" is used.
Summary: When solving problems, drawing and "one-to-one correspondence" methods can quickly find the answer.
2.Mathematical modeling.
Teacher: Think about it, if there are any other things in life that are similar to the problem of arranging flower pots, we can solve them with the method of "one-to-one correspondence".
Teachers and students communicated, and gradually presented: tree planting problems, street lamp problems, sawing problems, queuing problems, building climbing problems, etc.
Teacher: Think about it, who and who are "one-to-one" correspondings to these questions?
Talk to each other at the same table. Discuss in small groups, followed by a whole-class discussion, with the teacher using illustrations to help students understand. Student 1: We are talking about the problem of street lights, and the number of street lights corresponds to the number of intervals
Teacher: Think about it, what are the common characteristics of these problems, and they are all related to "spacing".
Teacher: Yes, whether it is the number of trees, the number of street lights, the number of people queuing, the number of floors, or the number of saws, they all correspond to the "number of intervals" and belong to the same type of mathematical problems. Mathematically, these problems are collectively referred to as "separation problems." Teacher: Do you think that in order to solve the problem of separation, the key is to find out what students find and find the number of intervals.
The teacher is right, I found the number of intervals, and then according to the method of one-to-one correspondence, you can find the number corresponding to it.
3. Application. 1.The garden team planted trees along one side of the 500-meter-long road (not at either end), planting one tree every 10 meters
2.A total of 15 street lights are installed on one side of the road (installed at both ends), and the distance between two adjacent street lights is 20 meters
Fourth, summary. Teacher: Students, think about what we have learned Xi today, and what have you gained?