Aggregation model. As the name implies, this model discusses the relationship between the total quantity and several partial quantities, some of which are equal and parallel, so the operation of this model needs addition. If we only consider it from the angle of mathematical calculation, we can also call this model addition model. This model can be specifically expressed as:
Total amount = partial amount+partial amount.
Obviously, this model can be used to solve a kind of problems involving total quantity in reality, which are common in mathematics teaching in lower grades of primary schools. For example, what is the total number of books in the library, what is the total cost of buying several goods in the store and so on. Furthermore, students can be guided to use the model flexibly according to the different backgrounds of specific problems in real life. For example, some stories can be described by "partial quantity", such as 14; You can also tell some stories about the total amount and turn the addition operation into the subtraction operation: partial amount.
= Total Amount-Partial Amount.
Distance model. This model tells us the relationship between distance, speed and time. If the speed is assumed to be uniform (or average speed), the form of the model can be obtained:
Distance = speed × time.
Although the distance problem is mentioned, this model can be applied to a kind of practical problems, such as "total price = unit price × quantity" and "total number = number of rows × number of columns".
Because this model emphasizes multiplication, it can also be called multiplication model from a purely mathematical point of view. Obviously, when using this kind of model, we can tell some stories at that time, such as how long A left later than B; You can also tell some stories about speed, for example, A changed the speed in the middle of the trip and so on. Of course, you can also tell some stories in the distance and turn multiplication into division: time = distance/speed.
According to different specific problems, the total model and the distance model can also be combined, and in the process of combination, the equation becomes a powerful mathematical tool. Through the construction and understanding of the model, we can gradually realize that mathematics is not only the abstraction of the relationship between quantity and graph in the real world, but also the model of logical reasoning. The concepts, methods and propositions formed by mathematics are also powerful tools to describe the real world.
In primary school mathematics teaching, there are two modes to be considered, one is tree planting mode, and the other is engineering mode, although there is no clear requirement in compulsory education mathematics curriculum standard.
Tree planting model. The problem background of this kind of model is: regularly dig some holes in a straight line or plane (or suppose there are some holes) and plant trees in the holes. Generally speaking, the number of trees planted is less than the number of holes, so two questions can be raised: one is to plant trees in some holes according to certain rules and ask how many trees can be planted; One kind of problem is to determine the number of trees planted and explore the law of tree planting. It is conceivable that in real life, such problems emerge one after another, which are also very interesting and meaningful. For example, setting up several gas stations along a road can regard the mileage of this road as a hole. For another example, if you want to set up several commercial points in an area, you can regard the residential area as a hole. Especially in modern society, this model is widely used in resource investigation or environmental investigation, because it is conceivable that there are several holes in the area to be investigated, and the investigation point is planting trees.
Obviously, it is much more difficult to design such a problem on a plane than on a straight line. Therefore, in primary school mathematics teaching, the background of the problem should be mainly aimed at straight lines rather than planes.
Engineering model. The problem background of this model is: there is a project, considering the time required for two engineering teams to complete the project together, it takes a day and b days for team A and team B to complete it alone. A simple way to solve this problem is to assume that the project is 1, because with this assumption, it can be determined that team A and team B can complete the project in one day: 1/A and 1/B respectively. Because of this, people also call such a problem a normalization problem. Of course, when using this model, it can be assumed that the cooperation between two engineering teams will improve efficiency or reduce efficiency; It can also be assumed that team A works for a few days before team B participates; It can also be assumed that there are three or more engineering teams to complete the project. The traditional problem of this model can also be the problem of water injection: several water pipes inject water into a pool, and you can also consider the situation of water injection and water discharge at the same time, and so on.
It can be seen that the process of using models can give full play to people's imagination. This kind of imagination is mainly manifested in constructing the realistic background, imagining various quantitative relations between things in the background, and imagining various possible combinations of quantitative relations. Therefore, in this teaching process, we should not only cultivate students' ability to analyze and solve problems, but also cultivate students' ability to find and ask questions. In fact, Example 54 in the compulsory education mathematics curriculum standard provides a good example. In this example, for the distance model, the quantitative relationship and some coordinate diagrams are given, so that students can judge the coordinate diagrams related to the quantitative relationship. In fact, it can also guide students to think about this kind of problems in turn, for example, first give a coordinate map and let students build a story about the distance model according to the quantitative relationship on the coordinate map.