With the progress of mankind, the development of science and technology, and the increasing digitalization of society, the application of mathematical modeling is more and more extensive, and the mathematical content around people is becoming more and more abundant. It is of great significance to emphasize the application of mathematics and cultivate the consciousness of applied mathematics to promote the implementation of quality education. The position of mathematical modeling in mathematical education has been promoted to a new height. Solving mathematical application problems through mathematical modeling can improve students' comprehensive quality. Based on the characteristics of mathematical application problems, this paper analyzes how to use mathematical modeling to solve mathematical application problems, hoping to get help and correction from colleagues.

First, the characteristics of mathematical application problems

We often call it a kind of mathematical problem, which comes from the reality of the objective world, has practical significance or background, and needs to be transformed into mathematical form through mathematical modeling, so as to be solved. Mathematical application problems have the following characteristics:

First, the mathematical application problem itself has practical significance or background. Reality here refers to all aspects of the real world, such as production reality, social reality, life reality and so on. For example, practical problems closely related to textbook knowledge and originated from real life; Application problems related to the intersection of modular subject knowledge networks; Application problems related to the development of modern science and technology, social market economy, environmental protection and realistic politics.

Secondly, the solution of mathematical application problems needs the method of mathematical modeling, which makes the problem mathematical, that is, the problem is transformed into mathematical form to express and then solved.

Third, there are many knowledge points involved in mathematical application problems. It is a test of the ability to comprehensively apply mathematical knowledge and methods to solve practical problems. It examines students' comprehensive ability and generally involves more than three knowledge points. If you don't master a certain knowledge point, it is difficult to answer the question correctly.

Fourthly, there is no fixed pattern or category for the proposition of mathematical application problems. It is often a novel practical background, which makes it difficult to train the problem model and solve the changeable practical problems with "sea tactics" Solving problems must rely on real ability, and the examination of comprehensive ability is more real and effective. Therefore, it has broad development space and potential.

Second, how to model mathematical application problems

Establishing mathematical model is the key to solving mathematical application problems. How to build a mathematical model can be divided into the following levels:

The first level: direct modeling.

According to the subject conditions, the ready-made mathematical formulas, theorems and other mathematical models are applied, and the explanatory diagram is as follows:

Conditional translation of themes

In mathematical expression,

Substitute the problem setting conditions of the application test into the mathematical model to solve.

Select the one that can be used directly.

mathematical model

The second level: direct modeling. You can use the existing mathematical model, but you must summarize this mathematical model, analyze the application problems, and then determine the specific mathematical model needed to solve the problems or the mathematical quantities needed in the mathematical model before you can use the existing mathematical model.

The third level: multiple modeling. Only by refining and dealing with complex relations, ignoring secondary factors and establishing several mathematical models can we solve the problem.

The fourth level: hypothesis modeling. Before the mathematical model is established, it needs to be analyzed, processed and assumed. For example, when we study the traffic flow at intersections, we can only model them when the traffic flow is stable and there are no emergencies.

Third, the ability to build mathematical models.

It is the key of the whole mathematics teaching process to establish a mathematical model from practical problems and solve practical problems by solving mathematical problems. Mathematical modeling ability is directly related to the quality of solving mathematical application problems, and also reflects a student's comprehensive ability.

3. 1 Improve analytical comprehension and reading ability.

Reading comprehension ability is the premise of mathematical modeling. Generally, mathematical application problems will create a new background, and some special terms will be used for the problem itself, and immediate definitions will be given. For example, 1999 college entrance examination question 22 gives the process description of cold-rolled steel strip, gives the special term "thinning rate" and gives a direct definition. Whether it can be understood deeply reflects its comprehensive quality, and this understanding ability directly affects the quality of mathematical modeling.

3.2 Strengthen the ability to transform written language narration into mathematical symbol language.

It is the basic work to translate all the words and images in mathematical application problems into mathematical symbolic language, that is, numbers, formulas, equations, inequalities, functions, etc.

For example, the original cost of a product is one yuan. In the next few years, it is planned to reduce the cost by p% on average every year compared with the previous year. What is the cost in five years?

The cost of translating the words given in the question into symbolic language is y=a( 1-p%)5.

3.3 Enhance the ability to choose mathematical models.

Choosing mathematical model is the embodiment of mathematical ability. There are many ways to establish mathematical models, and how to choose the best model to reflect the strength of mathematical ability. The establishment of mathematical models mainly involves equations, functions, inequalities, general term formulas of series, summation formulas, curve equations and other types. Combined with the teaching content, taking function modeling as an example, the following lists the mathematical models selected for practical problems:

Practical problems of function modeling types

Functions of cost, profit, sales revenue, etc.

Quadratic function optimization, material saving, minimum cost, maximum profit, etc.

Power function, exponential function, logarithmic function, cell division, biological reproduction, etc.

Trigonometric function measurement, alternating current, mechanical problems, etc.

3.4 Strengthen the ability of mathematical operation.

Mathematical application problems are generally complicated and have approximate calculation. Although some ideas are correct and the modeling is reasonable, it lacks computing power and will give up all efforts. Therefore, strengthening the reasoning ability of mathematical operations is the key to the correct solution of mathematical modeling. It is not advisable to ignore the cultivation of computing ability, especially computing ability, and only pay attention to the reasoning process without paying attention to the computing process.

Using mathematical modeling to solve mathematical application problems is very conducive to thinking from multiple angles, levels and sides, cultivating students' divergent thinking ability, and is an effective way to improve students' quality and implement quality education. At the same time, the application of mathematical modeling is also a kind of scientific practice, which is conducive to the cultivation of practical ability and a necessary condition for the implementation of quality education, and needs enough attention from educators.

Strengthening the teaching of mathematical modeling in senior high school and cultivating students' innovative ability

Based on the teaching of new mathematics textbooks in senior high school, combined with the compiling characteristics of new textbooks and the development of research-based learning in senior high school, this paper explores how to strengthen mathematics modeling teaching in senior high school and cultivate students' innovative ability.

Keywords: innovative ability; Mathematical modeling; Research study.

"Full-time senior high school mathematics syllabus (Trial)" puts forward new teaching requirements for students, requiring them to:

(1) Learn to ask questions and clarify the direction of inquiry;

(2) Experiencing the process of mathematical activities;

(3) Cultivate innovative spirit and application ability.

Among them, innovative consciousness and practical ability are one of the most prominent features in the new syllabus. Mathematics learning should not only cultivate and improve the basic knowledge of mathematics, basic skills and thinking ability, calculation ability and spatial imagination ability, but also cultivate and improve the ability of applying mathematics to analyze and solve practical problems. It is not enough to cultivate students' ability to analyze and solve practical problems only by classroom teaching. It is an important purpose and basic principle of mathematics teaching to have practice and cultivate students' innovative consciousness and practical ability. In order to make students learn to ask questions, make clear the direction of inquiry, communicate with existing knowledge and abstract practical problems into mathematical problems, it is necessary to establish mathematical models, thus forming a relatively complete mathematical knowledge structure.

Mathematical model is a bridge between mathematical knowledge and mathematical application. Studying and studying mathematical models can help students explore the application of mathematics, arouse their interest in mathematics learning, cultivate their innovative consciousness and practical ability, and strengthen the teaching of mathematical modeling, which is of far-reaching significance to the intellectual development of students. This paper talks about some experiences on how to strengthen the teaching of mathematical modeling in senior high school.

First, we should pay attention to the problem teaching before each chapter, so that students can understand the practical significance of establishing mathematical models.

Each chapter of the textbook is introduced by a related practical problem, which can directly tell students that after learning the teaching contents and methods of this chapter, this practical problem can be solved by mathematical model, so that students will have innovative consciousness, desire for new mathematical model and practical consciousness, and try it in practice after learning.

For example, in the new textbook Trigonometric Function, it is proposed that there is a semi-circular open space centered on point O, and an inscribed rectangle ABCD should be drawn on this open space to turn it into a green book, so that the edge AD of the book falls on the diameter of the semi-circle, and the other two points BC fall on the circumference of the semi-circle. Given that the radius of a semicircle is a, how to choose the positions of point A and point D symmetrical about point O to maximize the rectangular area?

This is a good opportunity to cultivate innovative consciousness and practical ability. We should pay attention to guidance, make an abstract analysis of the practical problems investigated, establish corresponding mathematical models, and put forward new knowledge through old and new ways of thinking, so as to stimulate students' desire for knowledge, such as not dampening students' enthusiasm and losing "bright spots".

In this way, through the problem teaching before the chapter, students can understand that mathematics is learning, researching and applying mathematical models, and at the same time cultivate their awareness of pursuing new methods and participating in practice. Therefore, we should attach importance to the teaching of the problems in the previous chapter, supplement some examples according to the needs of the construction and development of market economy and the problems found in students' practical activities, strengthen the teaching in this respect, make students pay attention to mathematics in their daily life and study, and cultivate their awareness of mathematical modeling.

2. The idea and thinking process of mathematical modeling permeate the teaching of solving application problems through geometry, triangle measurement problems and equations.

Learning the measurement of geometry and trigonometry can make students feel the idea of mathematical modeling in many ways, let them know more about the current mathematical model, consolidate the thinking process of mathematical modeling, and show the following modeling process to students in teaching:

Realistic prototype problem

mathematical model

Mathematical abstraction

Principle of simplification

Calculus reasoning

The Solution of Realistic Prototype Problem

Solution of mathematical model

Reflection principle

Return to explanation

Solving practical problems with equations embodies the idea that in the process of mathematical modeling thinking, problems should be deformed and simplified according to information and background materials to facilitate solving. And the important step in the process of solving problems is to solve equations according to the meaning of problems, so that students can understand that the key and difficult points in the process of mathematical modeling are through observation, analogy, induction, analysis and generalization, and construct new mathematical models to solve problems according to the characteristics of actual problems. Such as the series model of interest (compound interest), the equation model of profit calculation, the function model of decision-making problem and the inequality model.

3. Combining with the study of each chapter, cultivate students' ability to build mathematical models and expand the diversity and vividness of mathematical modeling forms.

The new senior high school syllabus requires that at least one research topic be arranged every semester, which is to cultivate students' mathematical modeling ability, such as "installment payment problem" in the chapter of "series" and "application of vectors in this chapter in the plane direction". At the same time, we can design profit investigation, negotiation, procurement, sales and other similar issues. The following research questions are designed.

According to the data given in the following table, the law of population growth in this country is determined and the population in 2000 is predicted.

Time (year)1910192019301940196019701980/.

Population (millions) 39 50 63 76 92106123132145.

Analysis: This is a question of determining the pattern of population growth. To simplify the problem, the following assumptions should be made: (1) The political, economic and social environment of the country is stable; (2) The population growth in this country is caused by the birth and death of the population; (3) Population quantification is continuous. Based on the above assumptions, we think that population is a function of time. The idea of modeling is to draw a scatter diagram according to the given data, and then find a straight line or curve to make them coincide with these scatter points as much as possible. This straight line or curve is considered as an approximate description of the law of population growth in this country, so as to make further predictions.

Through the study of the above problems, we not only reviewed and consolidated the knowledge of functions, but also cultivated students' mathematical modeling ability, practical ability and innovative consciousness. Pay attention to cultivating students to solve real life problems with mathematical models in daily teaching; Cultivate students' awareness of "number" and their ability to observe and practice in life, such as remembering some commonly used and common data, such as the speed of driving and cycling, height and weight. Make use of school conditions, organize students to practice on the playground, and once the activities are over, return to the classroom to solve practical problems into corresponding mathematical models. For example, the relationship between the angle and distance of shot put; The whole class hand in hand to form a rectangular circle, how to make the enclosed area the largest, and build dominoes with bricks.

Fourthly, cultivate students' other abilities and improve their thinking of mathematical modeling.

Because the thinking method of mathematical model almost runs through the whole process of mathematics learning in primary and secondary schools, the thinking method of mathematical model is cultivated by solving mathematical application problems in primary schools, establishing function expressions in secondary schools and analyzing trajectory equations in geometry. Mastering and using this method skillfully is the key to cultivate students' ability to analyze and solve problems with mathematics. I think this needs to cultivate students' following abilities in order to better improve their thinking in mathematical modeling:

(1) Ability to understand practical problems;

(2) Insight ability, that is, the ability to grasp the key points of the system;

(3) the ability to analyze problems abstractly;

(4) the ability of "translation", that is, the ability to express practical problems that have been abstracted and simplified for a lifetime with mathematical language symbols, to form mathematical models, and to express the ability to use mathematical methods to deduce or calculate results with natural language;

(5) Ability to use mathematical knowledge;

(6) Ability to pass the test of practice.

Only when the abilities in all aspects are strengthened can some knowledge be analogized, extrapolated and simplified. The following example requires all kinds of abilities to solve it smoothly.

Example 2: Solving Equation

x+y+z= 1 ( 1)

x2+y2+z2= 1/3 (2)

x3+y3+z3= 1/9 (3)

Analysis: If it is quite difficult to solve this problem with conventional solutions, it can be solved by carefully observing the conditions of the problem, mining hidden information, associating various knowledge, and constructing various equivalent mathematical models.

Equation model: Equation (1) represents the sum of three roots. From (1)(2), it is not difficult to get the sum of pairwise products (XY+YZ+ZX)= 1/3, and from (3), we can get the product of three roots (XYZ = 1/3). (4) X, Y and Z are just its three roots.

T3-T2+ 1/3t- 1/27 = 0(4)

Functional model:

According to (1)(2), if xz(x+y+z) is the coefficient of the first term and (x2+y2+z2) is a constant term, then 3 = (12+ 12) is the quadratic function f (x

Plane analysis model

Equation (1)(2) has a real number solution if and only if the straight line x+y= 1-z and the circle x2+y2= 1/3-z2 have a common point, and the necessary and sufficient condition for the latter to have a common point is the distance from the center of the circle (o, o) to the straight line x+y.

In a word, as long as teachers link mathematics knowledge with life and production practice by self-learning practical problems in teaching and according to the local and students' reality, students' awareness of applying mathematical models to solve practical problems can be enhanced, thus improving students' innovative consciousness and practical ability.

With the progress of mankind, the development of science and technology, and the increasing digitalization of society, the application of mathematical modeling is more and more extensive, and the mathematical content around people is becoming more and more abundant. It is of great significance to emphasize the application of mathematics and cultivate the consciousness of applied mathematics to promote the implementation of quality education. The position of mathematical modeling in mathematical education has been promoted to a new height. Solving mathematical application problems through mathematical modeling can improve students' comprehensive quality. Based on the characteristics of mathematical application problems, this paper analyzes how to use mathematical modeling to solve mathematical application problems, hoping to get help and correction from colleagues.

First, the characteristics of mathematical application problems

We often call it a kind of mathematical problem, which comes from the reality of the objective world, has practical significance or background, and needs to be transformed into mathematical form through mathematical modeling, so as to be solved. Mathematical application problems have the following characteristics:

First, the mathematical application problem itself has practical significance or background. Reality here refers to all aspects of the real world, such as production reality, social reality, life reality and so on. For example, practical problems closely related to textbook knowledge and originated from real life; Application problems related to the intersection of modular subject knowledge networks; Application problems related to the development of modern science and technology, social market economy, environmental protection and realistic politics.

Secondly, the solution of mathematical application problems needs the method of mathematical modeling, which makes the problem mathematical, that is, the problem is transformed into mathematical form to express and then solved.

Third, there are many knowledge points involved in mathematical application problems. It is a test of the ability to comprehensively apply mathematical knowledge and methods to solve practical problems. It examines students' comprehensive ability and generally involves more than three knowledge points. If you don't master a certain knowledge point, it is difficult to answer the question correctly.

Fourthly, there is no fixed pattern or category for the proposition of mathematical application problems. It is often a novel practical background, which makes it difficult to train the problem model and solve the changeable practical problems with "sea tactics" Solving problems must rely on real ability, and the examination of comprehensive ability is more real and effective. Therefore, it has broad development space and potential.

Second, how to model mathematical application problems

Establishing mathematical model is the key to solving mathematical application problems. How to build a mathematical model can be divided into the following levels:

The first level: direct modeling.

According to the subject conditions, the ready-made mathematical formulas, theorems and other mathematical models are applied, and the explanatory diagram is as follows:

Conditional translation of themes

In mathematical expression,

Substitute the problem setting conditions of the application test into the mathematical model to solve.

Select the one that can be used directly.

mathematical model

The second level: direct modeling. You can use the existing mathematical model, but you must summarize this mathematical model, analyze the application problems, and then determine the specific mathematical model needed to solve the problems or the mathematical quantities needed in the mathematical model before you can use the existing mathematical model.

The third level: multiple modeling. Only by refining and dealing with complex relations, ignoring secondary factors and establishing several mathematical models can we solve the problem.

The fourth level: hypothesis modeling. Before the mathematical model is established, it needs to be analyzed, processed and assumed. For example, when we study the traffic flow at intersections, we can only model them when the traffic flow is stable and there are no emergencies.

Third, the ability to build mathematical models.

It is the key of the whole mathematics teaching process to establish a mathematical model from practical problems and solve practical problems by solving mathematical problems. Mathematical modeling ability is directly related to the quality of solving mathematical application problems, and also reflects a student's comprehensive ability.

3. 1 Improve analytical comprehension and reading ability.

Reading comprehension ability is the premise of mathematical modeling. Generally, mathematical application problems will create a new background, and some special terms will be used for the problem itself, and immediate definitions will be given. For example, 1999 college entrance examination question 22 gives the process description of cold-rolled steel strip, gives the special term "thinning rate" and gives a direct definition. Whether it can be understood deeply reflects its comprehensive quality, and this understanding ability directly affects the quality of mathematical modeling.

3.2 Strengthen the ability to transform written language narration into mathematical symbol language.

It is the basic work to translate all the words and images in mathematical application problems into mathematical symbolic language, that is, numbers, formulas, equations, inequalities, functions, etc.

For example, the original cost of a product is one yuan. In the next few years, it is planned to reduce the cost by p% on average every year compared with the previous year. What is the cost in five years?

The cost of translating the words given in the question into symbolic language is y=a( 1-p%)5.

3.3 Enhance the ability to choose mathematical models.

Choosing mathematical model is the embodiment of mathematical ability. There are many ways to establish mathematical models, and how to choose the best model to reflect the strength of mathematical ability. The establishment of mathematical models mainly involves equations, functions, inequalities, general term formulas of series, summation formulas, curve equations and other types. Combined with the teaching content, taking function modeling as an example, the following lists the mathematical models selected for practical problems:

Practical problems of function modeling types

Functions of cost, profit, sales revenue, etc.

Quadratic function optimization, material saving, minimum cost, maximum profit, etc.

Power function, exponential function, logarithmic function, cell division, biological reproduction, etc.

Trigonometric function measurement, alternating current, mechanical problems, etc.

3.4 Strengthen the ability of mathematical operation.

Mathematical application problems are generally complicated and have approximate calculation. Although some ideas are correct and the modeling is reasonable, it lacks computing power and will give up all efforts. Therefore, strengthening the reasoning ability of mathematical operations is the key to the correct solution of mathematical modeling. It is not advisable to ignore the cultivation of computing ability, especially computing ability, and only pay attention to the reasoning process without paying attention to the computing process.

Using mathematical modeling to solve mathematical application problems is very conducive to thinking from multiple angles, levels and sides, cultivating students' divergent thinking ability, and is an effective way to improve students' quality and implement quality education. At the same time, the application of mathematical modeling is also a kind of scientific practice, which is conducive to the cultivation of practical ability and a necessary condition for the implementation of quality education, and needs enough attention from educators.