In recent years, not only the college entrance examination has produced application questions every year, but also the senior high school entrance examination has strengthened the investigation of application questions. These application problems focus on mathematical modeling and examine students' mathematical application ability. However, students' scoring rate on application questions is far lower than other questions. One of the reasons is that students lack mathematical modeling ability and awareness of applied mathematics. Therefore, middle school mathematics teachers should strengthen the teaching of mathematical modeling, improve students' mathematical modeling ability, and cultivate students' awareness of mathematical application and innovation. Based on teaching practice, this paper talks about some learning experiences of mathematical modeling teaching in junior high school.
1. Mathematical modeling is a brief representation of the process of establishing a mathematical model, which can be illustrated by the following block diagram:
practical problem
Abstraction, simplification and clarification of variables and parameters
According to a certain "law" or "law", a clear mathematical relationship between variables and parameters is established.
Solve mathematical problems analytically or approximately.
Interpretation and verification
Began to be used
cannot get/go through
get through
1. 1 Examining questions To build a mathematical model, we must first carefully examine the questions. The topics of practical problems are generally long, involving many nouns and concepts. We should carefully examine the topics patiently, thoroughly decompose the background of practical problems, and make clear the purpose of modeling. Find out the main known issues in the problem and try to master all kinds of information of the modeling object; Excavate the internal laws of practical problems, and make clear the required conclusions and the restrictive conditions of the required conclusions.
1.2 Simplify the problem according to the characteristics of the actual problem and the purpose of modeling. Grasp the main factors, abandon the secondary factors, and make assumptions with accurate language according to the quantitative relationship, connecting with mathematical knowledge and methods. 1.3 abstractly links the known conditions with the problems to be solved, appropriately introduces parameter variables or properly establishes coordinate systems, translates written language into mathematical language, and expresses quantitative relations in the form of mathematical expressions, graphs or tables, thus establishing mathematical models. Whether the mathematical model established by the above method conforms to reality and whether it is optimized in theory and method, after solving and analyzing the model, it is usually necessary to test the rationality of the model with actual phenomena and data.
2. Specific modeling and analysis methods.
① Relationship analysis method: by finding the quantitative relationship between key quantities, the mathematical model method of the problem is established.
(2) List analysis: the method of exploring the mathematical model of the problem through the list.
(3) Image analysis method: a method of establishing a mathematical model of the problem by analyzing the quantitative relationship in the image.
3. Master the basic mathematical model of common mathematical application problems.
In junior high school, the following mathematical models are usually established to solve application problems:
(1) Establish a geometric model.
② Establish an equation or inequality model.
③ Establish trigonometric function model.
④ Establish a functional model.
situation
Miss Wang attended a party. There are 40 people at the party. If every two people shake hands, how many times do the participants shake hands?
Example 2 Design an appropriate packaging method.
(1) has 4 tapes. How many kinds of packaging are there? Which way to save wrapping paper?
(2) If there are 8 tapes, which way to save wrapping paper?
Example 3 shows that,, and are nonnegative real numbers. Prove:
The first two problems obviously need to be analyzed by establishing geometric models, and the third problem is difficult to be solved by inequality deformation, but it is easy to be solved by establishing geometric models.
As shown below.
Example 4 Two factories, A and B, printed 200,000 and 250,000 copies of eighth-grade mathematics textbooks for use by A and B respectively. The number of students in A and B is 6.5438+0.7 million and 280,000 respectively. It is known that the freight charges from Factory A to A and B are 200 yuan/10,000 copies and 654.38+0.80 yuan/10,000 copies respectively. The freight from Factory B to A and B is 220 yuan/10,000 copies and 2 10 yuan/10,000 copies respectively. (1) Let the total freight be W yuan, and a factory will transport X million copies to a certain place. Try to write the functional relationship between W and X; (2) How to arrange the transit plan to minimize the total freight?
We have learned some measurement methods. Now please observe the tall objects in the school, such as teaching buildings, flagpoles, trees and so on. How to measure their height?
This problem obviously needs to be analyzed and solved by establishing trigonometric function model.
Dad is going to buy a new pair of sports shoes for Xiaoming, but he wants Xiaoming to figure out what size shoes to wear. Xiao Ming went home and measured the length of his mother's size 36 shoes, 23 cm, and his father's size 4/kloc-0 shoes, 25.5 cm. So what size shoes are you wearing with a length of 2 1.5 cm?
The more reasonable mathematical model of this problem is a linear function.
Example 71997165438+18 October, the TV was showing the spectacular scene of the closure of the Three Gorges Project. At 8: 55, the river closure began. At that time, the surface of Longkou was 40 meters wide and the water depth was 60 meters. 1 1: 50, and the width of announcer's report is 34.4 meters. 13: 00, the announcer reported that the water surface was 3 1 m wide. At this time, Xiao Ming, who is by the TV set, said that it is now possible to estimate the closing time in the afternoon, from 8: 55 to 1 1: 50, and the forward speed will decrease by 1.9 meters per hour, from11. It will take more than five hours from 1 in the afternoon, that is, it will not close until after 6 pm. But at 3: 28 pm, there was exciting news on TV: the river was successfully intercepted! Xiao Ming later realized that his estimation method was not good. Now, please design a more reasonable estimation method (build a more reasonable mathematical model) based on the above data to make your calculation result more realistic.
Modeling rationality analysis: There are two evaluation points for the modeling rationality of this topic.
(1) The backfill speed is measured by how many cubic meters of filler are per hour. In this way, whether a reasonable backfill speed calculation model can be established becomes the first evaluation point.
⑵ Pay attention to the gradual acceleration of backfilling speed: the larger the cross section of water, the deeper the water, and the more filler washed away during backfilling, the slower the corresponding advancing speed. On the contrary, the faster the speed. How to reasonably assume that the backfill speed in the model is getting faster and faster is the second evaluation point.
3. Experience in designing mathematical modeling teaching activities.
(1) Encourage students to actively participate and turn the teaching process into a student activity process more consciously.
Teachers should not only be "speakers" and "always correct guides", but should always play the following roles: role models-he not only demonstrates the correct start, but also demonstrates the wrong start and the thinking skills of "setting things right". Staff-give some suggestions to solve problems and provide reference information, but they can't make decisions instead of students. Inquirer-pretend not to know, ask the reasons, find loopholes, and urge students to make it clear, make it clear, and complete the progress. Arbitrator and Appreciator-Judge the value, significance, advantages and disadvantages of students' work, and encourage students to have creative ideas and practices.
(2) Pay attention to the actual level of students, step by step.
Mathematical modeling has a gradual learning and adaptation process for both teachers and students. When designing mathematical modeling activities, teachers should especially consider students' actual ability and level. The starting point should be low and the form should be conducive to more students' participation. At the beginning of teaching, we should consciously introduce the application background of knowledge while explaining the knowledge. Training should be combined with the key links of application, such as practical language and mathematical language, and solving application problems with equations and inequalities. Gradually expand to let students use the existing mathematical knowledge to explain some actual results, describe some actual phenomena, solve some relatively certain application problems in imitation, independently solve the mathematical application problems and modeling problems provided by teachers, and finally develop into the ability to independently discover and put forward some practical problems and solve them through mathematical modeling.
③ Pay attention to the teaching of knowledge generation and development.
Because the process of knowledge generation and development itself contains rich mathematical modeling ideas, teachers should not only pay attention to the analysis of the background of practical problems, the simplification of parameters and the consistency of assumptions, but also pay attention to the analysis of mathematical modeling principles and processes, the transformation and application of mathematical knowledge and methods, and can not only teach the results of mathematical modeling but ignore the process of mathematical modeling.
④ Pay attention to the "activity" of mathematical application and mathematical modeling.
The purpose of mathematical application and mathematical modeling is not only to expand students' extracurricular knowledge of mathematics, nor to solve some specific problems, but to cultivate students' application consciousness, mathematical ability and mathematical quality. Therefore, we should not follow the routine of teachers giving lectures and students imitating exercises, but should pay more attention to the process and participation, and more reflect the characteristics of activities.
refer to
[1] Mathematics Curriculum Standard for Full-time Compulsory Education
[2] Middle school mathematical modeling