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 raised to a new level.
Height, through mathematical modeling to solve mathematical application problems, improve students' comprehensive quality. This paper will combine the characteristics of mathematical application problems to solve how to use mathematical modeling.
The application of mathematics is analyzed, hoping to get help and correction from colleagues.
First, the characteristics of mathematical application problems
We often extract reality from the objective world, which has practical significance or realistic background, and transform the problem into mathematical form through mathematical modeling.
A mathematical problem that can be solved is called a mathematical application problem. Mathematical application problems have the following characteristics:
First, the mathematical application problem itself has practical significance or background. The reality here refers to the reality of production, society and life.
The reality in this regard. 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; Yuxian
On behalf of the development of science and technology, social market economy, environmental protection, real politics and other related application issues.
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, and it examines the students' synthesis.
Ability 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, and it is difficult to train the problem pattern.
Tactics can't solve the ever-changing practical problems. Solving problems must rely on real ability, and the examination of comprehensive ability is more real and effective. Indeed
There is 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 the mathematical model, analyze the application problems, and then determine what you need to solve.
The specific mathematical model or the mathematical quantities needed in the mathematical model need to be further solved before the existing mathematical model can be used.
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, if we study the traffic flow at intersections, suppose cars.
The traffic is stable, and there is no emergency to model.
Third, the ability to build mathematical models.
The key to the whole process of mathematics teaching is to establish mathematical models and numbers from practical problems and solve mathematical problems so as to solve practical problems.
The ability to learn modeling 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 produce a new background, and some special terms are used in the problem itself, and
Give a definition immediately. 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 the real-time measurement.
Whether we can deeply understand the meaning reflects a person's comprehensive quality and 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, through the letter
Taking digital 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 modeling is reasonable, they lack computing power and will go ahead.
Give up all your efforts. Therefore, strengthening the ability of mathematical operation and reasoning is the key to the correct solution of mathematical modeling, ignoring the cultivation of operational ability, especially computational ability, and only
It is not advisable to attach importance to the reasoning process but not to the calculation 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, thus improving
The quality of students is an effective way to implement quality education. At the same time, the application of mathematical modeling is also a scientific practice, which is conducive to the cultivation of practical ability and the improvement of implementation quality.
What is necessary for education requires educators to pay enough attention to it.
Strengthening the teaching of mathematical modeling in senior high school and cultivating students' innovative ability
Starting from the teaching of new mathematics textbooks in senior high schools, this paper puts forward some suggestions on how to strengthen mathematics modeling in senior high schools, combining with the compiling characteristics of new textbooks and the development of research-based learning in senior high schools.
Teaching and cultivating students' innovative ability are explored.
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 be trained and improved in basic mathematics knowledge, basic skills and thinking ability, computing ability and spatial imagination ability. It is also necessary to cultivate the ability of applying mathematical analysis and solving practical problems.
Practice has improved, but it is not enough to cultivate students' ability to analyze and solve practical problems only by classroom teaching. It is necessary to practice and cultivate students' innovative consciousness.
Practical ability is an important purpose and basic principle of mathematics teaching. Students should learn to ask questions, make clear the direction of inquiry, and be able to apply what they have learned.
In order to exchange 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 learning mathematical models can help students explore the application of mathematics and have a good understanding of mathematics learning.
Interest, cultivating students' innovative consciousness and practical ability, and strengthening the teaching and learning of mathematical modeling are of far-reaching significance to students' intellectual development. This paper discusses 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 content and methods of this chapter, this practical problem will
It can be solved by mathematical model. In this way, students will have a sense of innovation, a desire for new mathematical models and a sense of practice. After learning, they should try it in practice.
For example, the new textbook Trigonometric Function proposes that there is a semi-circular open space with O point as the center, and an inscribed rectangle ABCD should be drawn on this open space.
For a green book, the edge AD of the book falls on the diameter of the semicircle, and the other two points BC fall on the circumference of the semicircle. Given that the radius of a semicircle is a, how to choose the pair about point O?
The positions of point A and point D can 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 and establish the corresponding mathematical model.
And through the old and new ways of thinking, put forward new knowledge 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 new methods.
Consciousness of participating in practice. Therefore, we should not only attach importance to the teaching of the problems in the previous chapter, but also meet the needs of the construction and development of market economy and the problems found in students' practical activities.
Topics, add some examples, strengthen teaching in this area, let students attach importance to mathematics in their daily life and study, and cultivate students' 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.
Studying the measurement problems of geometry and trigonometry will make students feel the idea of mathematical modeling in many aspects and in all directions, and let students have more understanding and consolidation of the current mathematical model.
Mathematical modeling thinking process, in teaching to show students the following modeling process:
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
Using equations to solve practical problems reflects that in the process of mathematical modeling thinking, problems should be deformed and simplified according to information and background materials, thus
Ideas that are conducive to answering. An 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 the foundation.
According to the characteristics of practical problems, through observation, analogy, induction, analysis, generalization and other basic ideas, the existing mathematical models are associated or transformed into problems, and a new mathematical model is constructed.
To solve the problem. 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, just to cultivate students' mathematical modeling ability, such as "staging" in the chapter of "series"
The problem of payment ","The plane direction is the application of' chapter in chapter' vector in physics "and so on. At the same time, we can design profit investigation, negotiation, procurement, sales and other similar issues.
Title. 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.
Settings; (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 believe that population
Quantity 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 and let them kiss these scatter points as much as possible.
It is generally believed that straight lines or curves approximately describe the law of population growth in this country, and further predictions can be made.
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. Notes in daily teaching
Train students to use mathematical models to solve problems in real life; Cultivate students' sense of "number" in life and their ability to observe practice, such as memory.
Some commonly used and common data, such as: the speed of people driving and cycling, their own height and weight, etc. Make use of school conditions and organize students to practice in the playground.
Learning activities, return to the classroom as soon as the activities are over, and turn practical problems into corresponding mathematical models to solve. For example, the relationship between the angle and distance of shot put; The whole class.
Put the handle into 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, primary schools establish function expressions and
The trajectory equation in analytic geometry breeds the thinking method of mathematical model. Mastering and using this method skillfully is to cultivate students' ability to use mathematics to analyze problems.
The key to problem-solving ability, I think this requires students to cultivate the following abilities to better improve mathematical modeling thinking:
(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) "Translation" ability, that is, the ability to express abstract and simplified practical problems in a lifetime with mathematical language symbols, and to form mathematical models and correct them.
The ability to express results in natural language can be obtained by mathematical deduction or calculation;
(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 then from (3), the product of three roots can be obtained.
(XYZ= 1/27), a cubic equation model can be constructed from Vieta theorem. (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 term of the coefficient of the second term.
= (12+12+12) T2-2 (x+y+z) t+(x2+y2+z2) = (t-x) 2+(t-y) 2+(t-z) 2.
X = y = z = 1/3 obtained from (1) also applies to (3).
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 if and only if the center of the circle (o, o) is straight.
The distance of the straight line x+y is not greater than the radius.
In short, as long as teachers link mathematics knowledge with life and production practice through self-study in teaching and according to local and students' reality, they will
It can enhance students' awareness of applying mathematical models to solve practical problems, 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. Emphasize mathematics
It is of great significance to apply 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. Through mathematical modeling,
Solve mathematical application problems and improve students' comprehensive quality. This paper will combine the characteristics of mathematical application problems and analyze how to use mathematical modeling to solve mathematical application problems, hoping to get it.
Help and correction from colleagues.
First, the characteristics of mathematical application problems
We often regard the reality from the objective world as having practical significance or background, and we should transform the problem into mathematical form through mathematical modeling, so as to get a solution.
There is a kind of math problem called math application problem. 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.
International. 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; With the development of modern science and technology, the social market
Application problems related to economy, environmental protection and practical 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, and examines students' comprehensive ability, involving
There are generally 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 realistic background, so it is difficult to train the problem pattern and solve it with "sea tactics"
Solve ever-changing practical problems. 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.
The mathematical quantities needed in the model or mathematical model need to be further solved before the existing mathematical model can be used.
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, we study the traffic flow at intersections, assuming that the traffic flow is stable and there is no traffic flow.
Emergencies and so on can be modeled.
Third, the ability to build mathematical models.
The key to the whole process of mathematics teaching is to establish a mathematical model from practical problems and solve mathematical problems so as to solve practical problems.
It 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. such as
1999 Question 22 of the National College Entrance Examination describes the process of cold-rolled steel strip, gives the special term "thinning rate" and gives an immediate definition. Can you deeply understand and reflect on yourself?
Comprehensive quality, this understanding ability directly affects the quality of mathematical modeling.
3.2 Strengthen the ability to transform written language narration into mathematical symbol language.
The ability to translate all the words and images expressing quantitative relations in mathematical application problems into mathematical symbol languages, that is, numbers, formulas, equations, inequalities, functions, etc. , is a number.
Learn the basic work of modeling.
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. mathematical modelling
It 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, select the following practical problems.
List of selected mathematical models:
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. Some people, despite their correct thinking and reasonable modeling, have insufficient computing power and give up all their efforts before the meeting. So strengthen
The reasoning ability of mathematical operation is the key to the correct solution of mathematical modeling. It ignores the cultivation of computing ability, especially computing ability, and only attaches importance to the reasoning process and ignores the computing process.
It is not advisable to do so.
Using mathematical modeling to solve mathematical application problems is very beneficial for thinking problems from multiple angles, levels and sides, cultivating students' divergent thinking ability and improving students' quality and progress.
Effective ways of quality education. At the same time, the application of mathematical modeling is also a scientific practice, which is conducive to the cultivation of practical ability and a necessary condition for the implementation of quality education, and needs to be paid attention to by educators.
The author's enough attention.