Literature 1. Samarskiy AA, Mikhailov AP Mathematical modeling: Ideas. Methods. Examples. - M .: Nauka, Volkov E.A. Numerical methods. - M .: Nauka, Turchak LI Fundamentals of numerical methods. - M .: Nauka, Kopchenova NV, Maron IA Computational mathematics in examples and problems. - M .: Nauka, 1972.

A little history from the manipulation of objects to the manipulation of concepts about objects replacement of the studied object, process or phenomenon with a simpler and more accessible for research equivalent inability to take into account the entire set of factors that determine the properties and behavior of an object

Role of models The building is ugly, fragile or does not fit into the surrounding landscape Demonstration of circulatory systems in nature is inhuman Voltages, for example, in the wings, may be too great Assembling electrical circuits for measurements is uneconomical

Relationship of the model with the original Creating a model presupposes the preservation of some properties of the original, and in different models these properties may be different. The building made of cardboard is much smaller than the real one, but it allows you to judge its appearance; the poster makes the circulatory system understandable, although it has nothing to do with organs and tissues; the model of the aircraft does not fly, but the voltages in its body correspond to the flight conditions.

Why are models used? 1. A model is more accessible for research than a real object, 2. It is easier and cheaper to study a model than real objects, 3. some objects cannot be studied directly: it is not yet possible, for example, to build a device for thermonuclear fusion or conduct experiments in the interiors of stars, 4. experiments with the past are impossible, experiments with economics or social experiments are unacceptable

Purpose of models 1. Using the model, you can identify the most significant factors that form the properties of an object. Since the model reflects only some characteristics of the object - the original, then by varying the set of these characteristics as part of the model, it is possible to determine the degree of influence of certain factors on the adequacy of the model's behavior.

The model is needed: 1. In order to understand how a specific object is arranged: what is its structure, properties, laws of development and interaction with the outside world. 2. In order to learn how to manage an object or process and determine the best ways to manage for the given goals and criteria. 3. In order to predict the behavior of the object and assess the consequences of various methods and forms of impact on the object (meteorological models, models of biosphere development).

The property of a correct model A well-constructed, good model has a remarkable property: its study allows you to gain new knowledge about the object - the original, despite the fact that when creating the model, only some of the main characteristics of the original were used

Material modeling The model reproduces the basic geometric, physical, dynamic and functional characteristics of the object under study, when a real object is compared to its enlarged or reduced copy, which allows research in laboratory conditions with the subsequent transfer of the properties of the studied processes and phenomena from the model to the object based on the theory of similarity (planetarium, models of buildings and apparatus, etc.). In this case, the research process is closely related to the material impact on the model, that is, it consists of a natural experiment. Thus, material modeling is by nature an experimental method.

Types of ideal modeling Intuitive - modeling objects that do not lend themselves to formalization or do not need it. A person's life experience can be considered his intuitive model of the world around him Signed - modeling that uses sign transformations of various types as models: diagrams, graphs, drawings, formulas, etc. and contains a set of laws by which you can operate with model elements

Mathematical modeling The study of an object is carried out on the basis of a model formulated in the language of mathematics and investigated using certain mathematical methods. realizing these models with a computer

Classification mat. models By purpose: descriptive optimization simulation By the nature of the equations: linear nonlinear By taking into account changes in the system in time: dynamic static By the property of the domain of definition of arguments: continuous discrete By the nature of the process: deterministic stochastic

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A mathematical model is a mathematical representation of reality, one of the variants of a model, as a system, the study of which allows one to obtain information about some other system. The process of building and studying mathematical models is called mathematical modeling. All natural and social sciences that use the mathematical apparatus, in fact, are engaged in mathematical modeling: they replace the object of study with its mathematical model and then study the latter. The connection of the mathematical model with reality is carried out using a chain of hypotheses, idealizations and simplifications. With the help of mathematical methods, as a rule, an ideal object constructed at the stage of meaningful modeling is described. General information

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No definition can fully cover real-life mathematical modeling activities. Despite this, the definitions are useful in that they try to highlight the most significant features. According to Lyapunov, mathematical modeling is an indirect practical or theoretical study of an object, in which not the object of interest to us is directly studied, but some auxiliary artificial or natural system (model), which is in some objective correspondence with the object being cognized, capable of replacing it in certain respects and giving, in its investigation, ultimately, information about the modeled object itself. In other versions, the mathematical model is defined as a substitute object for the original object, which provides the study of some of the properties of the original, as "the" equivalent "of the object, reflecting in mathematical form its most important properties - the laws to which it obeys, the connections inherent in its constituent parts", as a system of equations, or arithmetic ratios, or geometric figures, or a combination of both, the study of which by means of mathematics should answer the questions posed about the properties of a certain set of properties of an object of the real world, as a set of mathematical ratios, equations, inequalities that describe the basic laws inherent in the studied process, object or system. Definitions

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The formal classification of models is based on the classification of the mathematical tools used. Often built in the form of dichotomies. For example, one of the popular sets of dichotomies: Linear or nonlinear models; Lumped or distributed systems; Deterministic or Stochastic; Static or dynamic; Discrete or continuous, and so on. Each constructed model is linear or nonlinear, deterministic or stochastic, ... Naturally, mixed types are also possible: in one respect, concentrated (in terms of parameters), in another, distributed models, etc. Formal classification of models

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Along with the formal classification, models differ in the way the object is represented: Structural or functional models. Structural models represent an object as a system with its own structure and mechanism of functioning. Functional models do not use such representations and only reflect the externally perceived behavior (functioning) of an object. In their extreme expression, they are also called "black box" models. Combined model types are also possible, sometimes referred to as "gray box" models. Mathematical models of complex systems can be divided into three types: Models of the black box type (phenomenological), Models of the gray box type (a mixture of phenomenological and mechanistic models), Models of the white box type (mechanistic, axiomatic). Schematic representation of black box, gray box, and white box models Classification by object representation

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Almost all authors describing the process of mathematical modeling indicate that at first a special ideal structure, a meaningful model, is being built. There is no established terminology here, and other authors call this ideal object a conceptual model, a speculative model, or a pre-model. In this case, the final mathematical construction is called a formal model or simply a mathematical model obtained as a result of formalizing a given meaningful model (pre-model). The construction of a meaningful model can be carried out using a set of ready-made idealizations, as in mechanics, where ideal springs, rigid bodies, ideal pendulums, elastic media, etc. provide ready-made structural elements for meaningful modeling. However, in areas of knowledge where there are no fully completed formalized theories (the cutting edge of physics, biology, economics, sociology, psychology, and most other areas), the creation of meaningful models becomes much more complicated. Content and formal models

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Peierls' work provides a classification of mathematical models used in physics and, more broadly, in natural sciences. In the book of A. N. Gorban and R. G. Khlebopros, this classification is analyzed and expanded. This classification is focused primarily on the stage of constructing a meaningful model. Hypothesis Models of the first type - hypotheses (“this could be”), “represent a trial description of a phenomenon, and the author either believes in its possibility, or even considers it to be true”. According to Peierls, these are, for example, Ptolemy's model of the solar system and Copernicus's model (improved by Kepler), Rutherford's model of the atom and the Big Bang model. Model-hypotheses in science cannot be proven once and for all, one can only talk about their refutation or non-refutation as a result of an experiment. If a model of the first type is built, then this means that it is temporarily recognized as true and it is possible to concentrate on other problems. However, this cannot be a point in research, but only a temporary pause: the status of a model of the first type can only be temporary. Phenomenological model The second type - the phenomenological model ("we behave as if ..."), contains a mechanism for describing the phenomenon, although this mechanism is not convincing enough, cannot be sufficiently confirmed by the available data, or does not agree well with the existing theories and accumulated knowledge about the object ... Therefore, phenomenological models have the status of temporary solutions. It is believed that the answer is still unknown and the search for "true mechanisms" must continue. Peierls refers, for example, to the second type, the caloric model and the quark model of elementary particles. The role of the model in research may change over time, it may happen that new data and theories confirm the phenomenological models and they will be promoted to the status of a hypothesis. Likewise, new knowledge may gradually come into conflict with hypothetical models of the first type, and those can be translated into the second. Substantial classification of models

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Thus, the quark model is gradually passing into the category of hypotheses; atomism in physics arose as a temporary solution, but with the course of history passed into the first type. But the ether models have made their way from type 1 to type 2, and now they are outside of science. The idea of ​​simplification is very popular when building models. But simplification is different. Peierls identifies three types of modeling simplifications. Approximation The third type of model is approximation (“something is considered very large or very small”). If it is possible to construct equations describing the system under study, this does not mean that they can be solved even with the help of a computer. The generally accepted technique in this case is the use of approximations (models of type 3). Among them are linear response models. Equations are replaced by linear ones. Ohm's law is a standard example. If we use the ideal gas model to describe sufficiently rarefied gases, then this is a type 3 (approximation) model. At higher gas densities, it is also useful to imagine a simpler situation with an ideal gas for qualitative understanding and assessments, but then it is already type 4. Simplification The fourth type is simplification ("for clarity, for clarity, some details"), in which details that can noticeably and not always controllable to influence the result. The same equations can serve as a model of type 3 (approximation) or 4 (we omit some details for clarity) - this depends on the phenomenon for which the model is used to study. So, if linear response models are used in the absence of more complex models (that is, nonlinear equations are not linearized, but linear equations describing the object are simply searched for), then these are already phenomenological linear models, and they belong to the following type 4 (all nonlinear details " we omit for clarity). Examples: application of the ideal gas model to the imperfect gas, the van der Waals equation of state, most models of solid state physics, liquids and nuclear physics. The path from microdescription to the properties of bodies (or media) consisting of a large number of particles, Substantive classification of models (continued)

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very long. Many details have to be discarded. This leads to the fourth type of model. Heuristic model The fifth type is a heuristic model ("there is no quantitative confirmation, but the model contributes to a deeper insight into the essence of the matter"), such a model retains only a qualitative semblance of reality and gives predictions only "in order of magnitude." A typical example is the mean free path approximation in kinetic theory. It gives simple formulas for the coefficients of viscosity, diffusion, thermal conductivity, consistent with reality in order of magnitude. But when building a new physics, it is far from immediately possible that a model is obtained that gives at least a qualitative description of an object - a model of the fifth type. In this case, a model is often used by analogy, reflecting reality at least in some way. Analogy The sixth type is an analogy model (“let's take into account only some of the features”). Peierls gives a history of using analogies in Heisenberg's first paper on the nature of nuclear forces. Thought experiment The seventh type of model is a thought experiment (“the main thing is to refute the possibility”). This type of modeling was often used by Einstein, in particular, one of such experiments led to the construction of the special theory of relativity. Suppose that in classical physics we are following a light wave at the speed of light. We will observe a periodically changing in space and constant in time electromagnetic field. According to Maxwell's equations, this cannot be. From this, Einstein concluded: either the laws of nature change when the frame of reference changes, or the speed of light does not depend on the frame of reference, and chose the second option. Demonstration of the possibility The eighth type is the demonstration of the possibility ("the main thing is to show the internal consistency of the possibility"), such models are also thought experiments with imaginary entities, demonstrating that the alleged phenomenon is consistent with the basic principles and Substantial classification of models (continued)

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internally consistent. This is the main difference from the Type 7 models, which reveal hidden contradictions. One of the most famous of these experiments is Lobachevsky's geometry. (Lobachevsky called it "imaginary geometry".) Another example is the mass production of formal kinetic models of chemical and biological oscillations, autowaves. The Einstein-Podolsky-Rosen paradox was conceived as a thought experiment to demonstrate the inconsistency of quantum mechanics, but in an unplanned way, over time, it turned into a type 8 model - a demonstration of the possibility of quantum teleportation of information. The substantive classification is based on the stages preceding mathematical analysis and calculations. The eight types of Peierls models are the eight types of research positions in modeling. Substantive classification of models (continued)

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virtually useless. Often, a simpler model allows for a better and deeper investigation of the real system than a more complex (and, formally, "more correct"). If we apply the harmonic oscillator model to objects that are far from physics, its meaningful status may be different. For example, when this model is applied to biological populations, it should most likely be classified as a type 6 analogy (“let's take into account only some of the features”). Example (continued)

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The most important mathematical models usually have an important property of universality: fundamentally different real phenomena can be described by the same mathematical model. For example, a harmonic oscillator describes not only the behavior of a load on a spring, but also other oscillatory processes, often of a completely different nature: small oscillations of a pendulum, oscillations of the liquid level in a U-shaped vessel, or a change in current strength in an oscillatory circuit. Thus, studying one mathematical model, we study at once a whole class of phenomena described by it. It is this isomorphism of laws expressed by mathematical models in various segments of scientific knowledge that Ludwig von Bertalanffy's feat to create a "general theory of systems". Versatility of models

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There are many problems associated with mathematical modeling. First, it is necessary to come up with the basic scheme of the modeled object, to reproduce it within the framework of the idealizations of this science. So, a train car turns into a system of plates and more complex bodies made of different materials, each material is set as its standard mechanical idealization (density, elastic moduli, standard strength characteristics), after which equations are drawn up, along the way some details are discarded as insignificant, calculations are made, compared with measurements, the model is refined, and so on. However, for the development of mathematical modeling technologies, it is useful to disassemble this process into its main constituent elements. Traditionally, there are two main classes of problems associated with mathematical models: direct and inverse. Direct task: the structure of the model and all its parameters are considered known, the main task is to conduct a study of the model to extract useful knowledge about the object. What static load will the bridge withstand? How it will react to a dynamic load (for example, on the march of a company of soldiers, or on the passage of a train at different speeds), how an airplane will overcome the sound barrier, whether it will fall apart from a flutter - these are typical examples of a direct task. Setting the correct direct problem (asking the correct question) requires special skill. If the right questions are not asked, the bridge can collapse, even if a good model has been built for its behavior. So, in 1879 in Great Britain, a metal railway bridge over the River Tay collapsed, the designers of which built a model of the bridge, calculated it for a 20-fold safety factor for the payload, but forgot about the constantly blowing winds in those places. And after a year and a half, it collapsed. In the simplest case (one oscillator equation, for example) the direct problem is very simple and reduces to an explicit solution of this equation. Inverse problem: many possible models are known, it is necessary to choose a specific model based on additional data Direct and inverse problems of mathematical modeling

"Systems approach to modeling" - Process - dynamic change of the system in time. System - a set of interrelated elements that form integrity or unity. Peter Ferdinand Drucker. Systematic approach in organizations. A systematic approach as the basis for introducing specialized training. Founders of the system approach: Structure is a way of interaction between the elements of the system through certain connections.

"ISO 20022" - Elements of the methodology of the international standard. Comparison of composition and properties. Appointment. Simulation process. Features of the methodology. Simulation results. Openness and development. Migration. Title of the International Standard. Aspects of versatility. Tools. Activity. Composition of documents.

"The concept of model and modeling" - Types of models by industry. Types of models. Basic concepts. Types of models depending on the time. Types of models depending on external dimensions. Adequacy of models. Figurative and iconic models. The need to create models. Modeling. Simulation modeling.

"Models and Modeling" - Changing the sizes and proportions. A mathematical model is a model presented in the language of mathematical relations. The block diagram is one of the special varieties of the graph .. Analysis of the object. Structural model - representation of an informational sign model in the form of a structure. A real phenomenon. Abstract. Verbal.

Model Development Steps - Descriptive information models are usually built using natural languages ​​and pictures. Building a descriptive information model. The main stages of development and research of models on a computer. Stage 4. Stage 1. Stage 5. Solar system model. Practical task. Stage 3. Stage 2.

"Modeling as a method of cognition" - In biology - the classification of the animal world. Definitions. Definition. In physics, it is an information model of simple mechanisms. Modeling as a method of cognition. Forms of presentation of information models. Tabular model. The process of building information models using formal languages ​​is called formalization.

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Presentation on the topic: Mathematical models (grade 7)

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§ 2.4. Mathematical Models The main language of information modeling in science is the language of mathematics. Models built using mathematical concepts and formulas are called mathematical models. A mathematical model is an information model in which parameters and relationships between them are expressed in mathematical form.

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Mathematical modeling The modeling method makes it possible to apply the mathematical apparatus to solving practical problems. The concepts of numbers, geometrical figures, equations, are examples of mathematical models. The method of mathematical modeling in the educational process has to be resorted to when solving any problem with practical content. To solve such a problem by mathematical means, it must first be translated into the language of mathematics (to build a mathematical model).

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In mathematical modeling, the study of an object is carried out by studying a model formulated in the language of mathematics, for example: you need to determine the surface area of ​​a table. Measure the length and width of the table, and then multiply the resulting numbers. This effectively means that the real object - the table surface - is replaced by an abstract mathematical model with a rectangle. The area of ​​this rectangle is considered to be the desired one. Of all the properties of the table, three were distinguished: the shape of the surface (rectangle) and the lengths of the two sides. Neither the color of the table, nor the material from which it is made, nor how it is used are important. Assuming the table surface is a rectangle, it's easy to specify the input and output. They are related by the relation S = ab.

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Let's consider an example of reducing the solution of a specific problem to a mathematical model. Through the porthole of the sunken ship, you need to pull out a chest with jewels. Some assumptions about the shape of the chest and porthole windows and the initial data for solving the problem are given. Assumptions: The porthole is circular. The chest has the shape of a rectangular parallelepiped. Initial data: D - window diameter; x is the length of the chest; y is the width of the chest; z is the height of the chest. End result: Message: Can or cannot be pulled.

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A systematic analysis of the problem statement revealed the relationship between the size of the window and the size of the chest, taking into account their shape. The information obtained as a result of the analysis was displayed in the formulas and relationships between them, so a mathematical model arose. The mathematical model for solving this problem is the following dependencies between the initial data and the result:

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Example 1: Calculate the amount of paint to cover the floor in a gym. To solve the problem, you need to know the floor area. To complete this task, measure the length, width of the floor and calculate its area. The real object - the floor of the hall - is occupied by a rectangle, for which the area is the product of length and width. When buying paint, they find out how much area can be covered with the contents of one can, and calculate the required number of cans: Let A - floor length, B - floor width, S1 - area that can be covered with the contents of one can, N - the number of cans. The floor area is calculated using the formula S = A × B, and the number of cans required to paint the hall is N = A × B / S1.

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Example 2: Through the first pipe, the pool is filled in 30 hours, through the second pipe - in 20 hours. How many hours will it take for the pool to be filled through two pipes? Solution: Let us denote the time of filling the pool through the first and second pipes A and B, respectively. We take the entire volume of the pool as 1, and denote the required time by t. Since through the first pipe the pool is filled in A hours, then 1 / A is the part of the pool filled with the first pipe in 1 hour; 1 / B - the part of the pool filled with the second pipe in 1 hour. Therefore, the rate of filling the pool with the first and second pipes together will be: 1 / A + 1 / B. You can write down: (1 / A + 1 / B) t = 1. obtained a mathematical model describing the process of filling a pool from two pipes. The required time can be calculated by the formula:

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Example 3: Points A and B are located on the highway, which are 20 km apart. The motorcyclist left point B in the direction opposite to A at a speed of 50 km / h. Let us compose a mathematical model describing the position of the motorcyclist relative to point A in t hours. In t hours the motorcyclist will travel 50t km and will be from A at a distance of 50t km + 20 km ... If we denote by the letter s the distance (in kilometers) of the motorcyclist to point A, then the dependence of this distance on the time of movement can be expressed by the formula: S = 50t + 20, where t> 0. The mathematical model for solving this problem is the following dependencies between the initial data and the result: Misha had x marks; Andrey has 1.5x. Misha's got x-8, Andrey's 1.5x + 8. By the condition of the problem, 1.5x + 8 = 2 (x-8).

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The mathematical model for solving this problem is the following relationship between the initial data and the result: Misha had x brands; Andrey has 1.5x. Misha's got x-8, Andrey's 1.5x + 8. By the condition of the problem, 1.5x + 8 = 2 (x-8). The mathematical model for solving this problem is the following dependencies between the initial data and the result: x people work in the second shop, 4x in the first, and x + 50 in the third. x + 4x + x + 50 = 470. The mathematical model for solving this problem is the following dependencies between the initial data and the result: the first number x; second x + 2.5. By the condition of the problem, x / 5 = (x + 2.5) / 4.

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Sources Informatics and ICT: a textbook for grade 7 Author: Bosova L. L. Publisher: BINOM. Knowledge Laboratory, 2009 Format: 60x90 / 16 (per.), 229 p., ISBN: 978-5-9963-0092-1http: //www.lit.msu.ru/ru/new/study (graphs, diagrams ) http://images.yandex.ru (pictures)

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Mathematical models

05.05.17 Mathematical Models The main language of information modeling in science is the language of mathematics. Models built using mathematical concepts and formulas are called mathematical models. Mathematical model is an informational model in which parameters and relationships between them are expressed in mathematical form.

05.05.17 For example, the well-known equation S = vt, where S - distance, v - speed t - time, is a model of uniform motion, expressed in mathematical form.

05.05.17 Considering the physical system: a body of mass m, rolling down an inclined plane with acceleration a under the action of a force F, Newton obtained the relation F = ma. It is a mathematical model of a physical system.

05.05.17 The modeling method makes it possible to apply the mathematical apparatus to solving practical problems. The concepts of numbers, geometrical figures, equations, are examples of mathematical models. The method of mathematical modeling in the educational process has to be resorted to when solving any problem with practical content. To solve such a problem by mathematical means, it must first be translated into the language of mathematics (to build a mathematical model). Math modeling

05.05.17 In mathematical modeling, the study of an object is carried out by studying a model formulated in the language of mathematics. Example: you need to determine the surface area of ​​the table. Measure the length and width of the table, and then multiply the resulting numbers. This effectively means that the real object - the table surface - is replaced by an abstract mathematical model with a rectangle. The area of ​​this rectangle is considered to be the desired one. Of all the properties of the table, three were distinguished: the shape of the surface (rectangle) and the lengths of the two sides. Neither the color of the table, nor the material from which it is made, nor how it is used are important. Assuming the table surface is a rectangle, it's easy to specify the input and output. They are related by the relation S = ab.

05/05/17 Let's consider an example of reducing the solution of a specific problem to a mathematical model. Through the porthole of the sunken ship, you need to pull out a chest with jewels. Some assumptions about the shape of the chest and porthole windows and the initial data for solving the problem are given. Assumptions: The porthole is circular. The chest has the shape of a rectangular parallelepiped. Initial data: D - window diameter; x is the length of the chest; y is the width of the chest; z is the height of the chest. End result: Message: Can or cannot be pulled.

05/05/17 If, then the chest can be pulled out, and if, then it is impossible. A systematic analysis of the problem statement revealed the relationship between the size of the porthole and the size of the chest, taking into account their shape. The information obtained as a result of the analysis was displayed in the formulas and relationships between them, so a mathematical model arose. The mathematical model for solving this problem is the following dependencies between the initial data and the result:

05/05/17 Example 1: Calculate the amount of paint to cover the floor in a gym. To solve the problem, you need to know the floor area. To complete this task, measure the length, width of the floor and calculate its area. The real object - the floor of the hall - is occupied by a rectangle, for which the area is the product of length and width. When buying paint, they find out how much area can be covered with the contents of one can, and calculate the required number of cans. Let A be the length of the floor, B the width of the floor, S 1 the area that can be covered with the contents of one can, N the number of cans. The floor area is calculated using the formula S = A × B, and the number of cans required to paint the hall is N = A × B / S 1.

05.05.17 Example 2: The pool is filled through the first pipe in 30 hours, through the second pipe - in 20 hours. How many hours will the pool fill through two pipes? Solution: Let's designate the filling time of the pool through the first and second pipes A and B, respectively. We take the entire volume of the pool as 1, and denote the required time by t. Since through the first pipe the pool is filled in A hours, then 1 / A is the part of the pool filled with the first pipe in 1 hour; 1 / B - part of the pool filled with the second pipe in 1 hour. Therefore, the rate of filling the pool with the first and second pipes together will be: 1 / A + 1 / B. You can write: (1 / A + 1 / B) t = 1. obtained a mathematical model describing the process of filling a pool from two pipes. The required time can be calculated by the formula:

05/05/17 Example 3: Points A and B are located on the highway, located at a distance of 20 km from each other. The motorcyclist left point B in the opposite direction to A at a speed of 50 km / h. Let's compose a mathematical model describing the position of the motorcyclist relative to point A in t hours. In t hours, the motorcyclist will travel 50 t km and will be from A at a distance of 50 t km + 20 km. If we denote the distance (in kilometers) of the motorcyclist to point A with the letter s, then the dependence of this distance on the time of movement can be expressed by the formula: S = 50t + 20, where t> 0.

05/05/17 The first number is equal to x, and the second is 2.5 more than the first. It is known that 1/5 of the first number is equal to 1/4 of the second. Make mathematical models of these situations: Misha has x marks, and Andrey has one and a half times more. If Misha gives Andrey 8 marks, then Andrey will have twice as many marks as Misha will have. The second workshop employs x people, in the first - 4 times more than in the second, and in the third - 50 more people than in the second. In total, 470 people are employed in three workshops of the plant. Let's check: The following dependencies between the initial data and the result are the mathematical model for solving this problem: Misha had marks; Andrey has 1.5x. Misha's got x-8, Andrey's 1.5x + 8. By the condition of the problem, 1.5x + 8 = 2 (x-8). The mathematical model for solving this problem is the following dependencies between the initial data and the result: x people work in the second shop, 4x in the first, and x + 50 in the third. x + 4x + x + 50 = 470. The mathematical model for solving this problem is the following dependencies between the initial data and the result: the first number x; second x + 2.5. By the condition of the problem, x / 5 = (x + 2.5) / 4.

05/05/17 This is how mathematics is usually applied to real life. Mathematical models are not only algebraic (in the form of equality with variables, as in the examples discussed above), but also in another form: tabular, graphical, and others. We will get acquainted with other types of models in the next lesson.

05/05/17 Assignment at home: § 9 (pp. 54-58) No., 2, 4 (p. 60) in the notebook

05/05/17 Thanks for the lesson!

05.05.17 Sources Informatics and ICT: textbook for grade 8 http://www.lit.msu.ru/ru/new/study (graphs, diagrams) http://images.yandex.ru (pictures)