>>Informatics: Representation of dependencies between quantities
Representation of dependencies between quantities
Solving planning and management problems constantly requires taking into account the dependencies of some factors on others.
Examples of dependencies:
1) the time a body falls to the ground depends on the initial height;
2) the pressure depends on the temperature of the gas in the cylinder;
Mathematical model- is a collection quantitative characteristics some object (process) and connections between them, presented in the language of mathematics.
Mathematical models for the first two examples listed above are well known. They reflect physical laws and are presented in the form of formulas:
These are examples of dependencies represented in a sawtooth function. The first dependency is called the root dependency (time is proportional to square root from height), the second - linear (pressure is directly proportional to temperature).
In more complex tasks mathematical models are represented in the form of equations or systems of equations. In this case, to extract functional dependence quantities you need to be able to solve these equations. At the end of this chapter, we will consider an example of a mathematical model that is expressed by a system of inequalities.
Let's look at examples of two other ways to present dependencies between quantities: tabular and graphical.
Imagine that we decided to test the law free fall bodies experimentally. The experiment was organized in the following way; throw a steel ball from the balcony of the 2nd floor, 3rd floor (and so on) of a ten-story building, measuring the height initial position ball and falling time. Based on the results of the experiment, we compiled a table and drew a graph.
"
Rice. 2.11. Tabular and graphical representation dependence of the time of falling of a body on height
If each pair of values of H and t from this table is substituted into the above formula for the dependence of height on time, then it will turn into an equality (to within the measurement error). This means the model works well. (However, if you throw not a steel ball, but big light ball then this model will correspond less to the formula, and if it is an inflatable ball, it will not correspond at all - why do you think?)
In this example, we looked at three ways to display the dependence of quantities: functional (formula), tabular and graphical. However, only a formula can be called a mathematical model of the process of a body falling to the ground. Why? Because the formula is universal. It allows you to determine the time of a body falling from any height, and not just for the experimental set of H values shown in Fig. 2.11.
In addition, the table and diagram(graph) state the facts, and mathematical model allows you to predict, predict through calculations.
In the same way, you can display the dependence of pressure on temperature in three ways. Both examples are related to known physical laws - the laws of nature. Knowledge physical laws allow to produce accurate calculations, they form the basis of modern technology.
Briefly about the main thing
Magnitude is some quantitative characteristic of an object.
Dependencies between quantities can be presented in the form of a mathematical model, in tabular and graphical forms.
The relationship, presented in the form of a formula, is a mathematical model.
Questions and tasks
1. a) What forms of representation of dependencies between quantities do you know?
b) What is a mathematical model?
c) Can a mathematical model include only constants?
2. Give an example of a functional relationship (formula) known to you between the characteristics of a certain system.
3. Justify the advantages and disadvantages of each of the three forms of representing dependencies.
Semakin I.G., Henner E.K., Computer Science and ICT, 11
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Lesson objectives:
1. Get acquainted with the concepts:
"magnitude"
"mathematical model",
"tabular model"
"graphical model"
Educational:
Create conditions for the development of the ability to highlight the main thing, compare, analyze, generalize.
Educational:
Cultivate attentiveness, the desire to bring the matter to the intended result;
Establishing mutual contacts and sharing experiences between students and the teacher.
Equipment: teacher's computer with multimedia projector.
Lesson Plan
Organizational moment (2 min) Setting lesson goals. Explanation of new material. (17 min) Reinforcing new material (5 min) Solving tasks from demo versions of the Unified State Exam 2010 (15 min) Summing up (3 min) Homework (3 min)
During the classes
Tell students the topic of the lesson. (slide 1) Setting a lesson goal(slide 2)
Lesson objectives:
1. Get acquainted with the concepts:
"magnitude"
"dependencies between quantities"
"mathematical model",
"tabular model"
"graphical model"
Consider the dependencies between quantities using examples.
2. Improve skills in solving tasks from the Unified State Exam KIMs.
Explanation of new material. (17 min)(slide 3)
Application mathematical modeling constantly requires taking into account the dependencies of some quantities on others.
1. The time a body falls to the ground depends on the initial height;
2. The gas pressure in the cylinder depends on its temperature;
3. Frequency of illnesses among residents bronchial asthma depends on the quality of urban air
(slide 4)
Any research must begin by identifying the quantitative characteristics of the object under study. Such characteristics are called quantities. There are three main properties associated with any quantity: name, values, type.
The name of the quantity can be full (gas pressure), or it can be symbolic (P). For certain quantities, standard names are used: time - T, speed - V, force - F...
(slide 5)
If the value of a quantity does not change, then it is called constant value or constant
(π =3.14159…).
A quantity that changes its value is called variable.
(slide 6)
A type defines the set of values that a value can take. Basic types of values: numeric, symbolic, logical. Since we will only talk about quantitative characteristics, we will only consider quantities numeric type.
(Slide 7)
Let's return to the examples and denote variables, the dependencies between which we are interested in.
In example 1:
T (sec) – fall time; N (m) – fall height. Gravity acceleration g (m/sec2) – constant.
In example 2: P(n/m2) – gas pressure ; t° C is the gas temperature.
IN example 3:
Air pollution is characterized by the concentration of impurities C (mg/cubic m). The incidence rate is characterized by the number of chronic asthma patients per 1000 inhabitants of this city– P(bol/thousand)
(Slide 8)
Let's look at Dependency Representation Methods
Mathematical model Tabular model Graphic model
(Slide 9)
Mathematical model
This is a set of quantitative characteristics of some object (process) and connections between them, presented in the language of mathematics.
For the first example, the mathematical model is presented as a formula:
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(Slide 11)
Graphic model
and draw a graph
 |
(Slide 12)
Information models that describe the development of systems over time have a special name: dynamic models.
IN physics dynamic information models describe the movement of bodies; V biology – development of organisms and animal populations; in chemistry – leakage chemical reactions etc
(slide 13)
The solution of the problem: (1 student at the blackboard, the rest in notebooks)
Build mathematical, tabular and graphical models of the problem:
The body moves according to the lawx (t)=5t2+2t-5,
Wherex – movement in meters,t – time in seconds. Find the speed of the body at the moment of timet=2.
Construct a table showing the dependence of the speed of a body on the time of movement of the body with an interval of 3 seconds.
Consolidation of the studied material.Answer the questions:
1. What forms of representation of dependencies between quantities do you know? (answer 1 student)
2. Justify the advantages and disadvantages of each three forms representation
dependencies. (answer 1 student)
Solving tasks from the demo version of the Unified State Exam 2010 (15 min)
Repetition of the 10th, 2nd, 8th and 16th number systems.
Solving the task from the demo version of the Unified State Exam (1 )
1. How is the number 26310 represented in the octal number system?
Solution:
How to write the number 5678 in binary system dead reckoning?
(1 student at the blackboard, the rest in notebooks)
Solution:
How is the number A8716 written in the octal number system?
(1 student at the blackboard, the rest in notebooks)
Solution:
Task A1 from the 2010 demo version. (1 student at the blackboard, the rest in notebooks)
Given: a=9D16, b=2378. Which of the numbers C, written in the binary number system, satisfies the inequality
Solution:
Summing up (3 min) Homework (3 min) §36, questions. Example.
Given: a= 3328, b= D416. Which of the numbers C, written in the binary number system, satisfies the inequality a Quantities and dependencies between them These are examples of dependencies represented in functional form. The first dependence is called root (time is proportional to the square root of the height), the second is linear. Modeling dependencies between quantities Value - quantitative characteristics of the object under study Quantity characteristics Meaning reflects the meaning of the quantity determines the possible values of the quantity constant Types of dependencies: Functional Methods for displaying dependencies Mathematical Tabular model Graphic Description of the development of systems over time - dynamic model The two quantities are called directly proportional, if when one of them increases several times, the other increases by the same amount. Accordingly, when one of them decreases several times, the other decreases by the same amount. The relationship between such quantities is a direct proportional relationship. Examples of direct proportional dependence: 1) at a constant speed, the distance traveled is directly proportional to time; 2) the perimeter of a square and its side are directly proportional quantities; 3) the cost of a product purchased at one price is directly proportional to its quantity. To distinguish a direct proportional relationship from an inverse one, you can use the proverb: “The further into the forest, the more firewood.” It is convenient to solve problems involving directly proportional quantities using proportions. 1) To make 10 parts you need 3.5 kg of metal. How much metal will go into making 12 of these parts? (We reason like this: 1. In the filled column, place an arrow in the direction from the largest number to the smallest. 2. The more parts, the more metal needed to make them. This means that this is a directly proportional relationship. Let x kg of metal be needed to make 12 parts. We make up the proportion (in the direction from the beginning of the arrow to its end): 12:10=x:3.5 To find , you need to divide the product of the extreme terms by the known middle term: This means that 4.2 kg of metal will be required. Answer: 4.2 kg. 2) For 15 meters of fabric they paid 1680 rubles. How much does 12 meters of such fabric cost? (1. In the filled column, place an arrow in the direction from the largest number to the smallest. 2. The less fabric you buy, the less you have to pay for it. This means that this is a directly proportional relationship. 3. Therefore, the second arrow is in the same direction as the first). Let x rubles cost 12 meters of fabric. We make a proportion (from the beginning of the arrow to its end): 15:12=1680:x To find the unknown extreme term of the proportion, divide the product of the middle terms by the known extreme term of the proportion: This means that 12 meters cost 1344 rubles. Answer: 1344 rubles. MODELING DEPENDENCIES BETWEEN VARIABLES INFORMATION MODELING TECHNOLOGIES Key Concepts Application of mathematical modeling The use of mathematical modeling constantly requires taking into account the dependencies of some quantities on others. Examples of dependencies:
Implementation mathematical model requires knowledge of techniques for representing dependencies between quantities. Dependency representation methods Magnitude– quantitative characteristics of the object under study Quantity characteristics reflects the meaning of the quantity determines the possible values of the quantity Meaning constant variable Main types of quantities: An example of a constant is the Pythagorean number The value name can be semantic semantic numerical "gas pressure" In describing the process of a body falling variable quantities are height H
and time of fall t
symbolic symbolic logical Types of dependencies Functional dependence
is a relationship between two quantities in which a change in one of them causes a change in the other. Example 1:
t(c) – fall time; H(m) – fall height. We will represent the dependence, neglecting air resistance; the acceleration of free fall g (m/s 2) will be considered a constant. Example 2:
P(n/m 2) – gas pressure (in SI units, pressure is measured in newtons per square meter); t°C – gas temperature. Pressure at zero degrees P We will consider 0 a constant for a given gas. certain . Types of dependencies Other addiction
is more complex in nature, the same value can take on different values, since it can be influenced by other indicators. Example 3:
Air pollution is characterized by the concentration of impurities – C (mg/m3). The unit of measurement is the mass of impurities contained in 1 cubic meter of air, expressed in milligrams. The incidence rate will be characterized by the number of chronic asthma patients per 1000 residents of a given city P(bol/thousand) The relationship between quantities is completely certain . Mathematical models Mathematical models - this is a set of quantitative characteristics of some object (process) and connections between them, presented in the language of mathematics. Mathematical models reflect physical laws and are presented in the form of formulas: Linear dependence Root dependence (time is proportional to the square root of the height) In complex problems, mathematical models are represented as equations or systems of equations. Tabular and graphical models Let's experimentally check the law of free fall of a body Experiment:
a steel ball is dropped from a 6-meter, 9-meter height, etc. (after 3 meters), measuring the height of the initial position of the ball and the time of fall The result of the experiment is presented in the table and graph N
, m t
, c Tabular and graphical representation of the dependence of the time of falling of a body on height Dynamic models Information models that describe the development of systems over time have a special name: dynamic models . In physics this is the movement of bodies, in biology - the development of organisms or animal populations, in chemistry – the occurrence of chemical reactions. The most basic Name – reflects the meaning of the quantity Type – defines possible values of quantities Value: constant value (constant) or variable Questions and tasks Illustrations: SourcesComputer Science and ICT grades 10-11 Semakin, Computer Science grades 10-11 Semakin, Modeling dependencies between quantities, Quantities and dependencies between them, Various methods of representing dependencies, Mathematical models, Tabular and graphical models
The content of this section of the textbook is related to computer mathematical modeling. The use of mathematical modeling constantly requires taking into account the dependencies of some quantities on others. Here are examples of such dependencies:
1) the time a body falls to the ground depends on its initial height;
2) the gas pressure in the cylinder depends on its temperature;
3) the level of morbidity of city residents with bronchial asthma depends on the concentration of harmful impurities in the city air.
Implementation of a mathematical model on a computer (computer mathematical model) requires knowledge of techniques for representing dependencies between quantities.
Let's look at different methods of representing dependencies.
Any research must begin by identifying the quantitative characteristics of the object under study. Such characteristics are called quantities.
You have already encountered the concept of quantity in the basic computer science course. Let us recall that three basic properties are associated with any quantity: name, value, type.
The name of a quantity can be semantic or symbolic. An example of a semantic name is “gas pressure,” and a symbolic name for the same quantity is P. In databases, quantities are record fields. As a rule, meaningful names are used for them, for example: SURNAME, WEIGHT, ASSESSMENT, etc. In physics and other sciences that use mathematical apparatus, symbolic names are used to denote quantities. To ensure that the meaning is not lost, standard names are used for certain quantities. For example, time is denoted by the letter t, speed by V, force by F, etc.
If the value of a quantity does not change, then it is called a constant quantity or constant. An example of a constant is the Pythagorean number π = 3.14259... . A quantity whose value can change is called a variable. For example, in the description of the process of falling of a body, the variable quantities are the height H and the falling time t.
The third property of a quantity is its type. You also came across the concept of a value type when learning about programming and databases. A type defines the set of values that a value can take. Basic types of values: numeric, symbolic, logical. Since in this section we will talk only about quantitative characteristics, then only quantities of a numerical type will be considered.
Now let’s return to examples 1-3 and denote (name) all the variable quantities, the dependencies between which will interest us. In addition to the names, we indicate the dimensions of the quantities. Dimensions define the units in which the values of quantities are represented.
1) t (s) — fall time; N (m) — fall height. We will represent the dependence, neglecting air resistance; the acceleration of free fall g (m/s 2) will be considered a constant.
2) P (n/m2) - gas pressure (in SI units, pressure is measured in newtons per square meter); t °С is the gas temperature. We will consider the pressure at zero degrees Po to be a constant for a given gas.
3) Air pollution will be characterized by the concentration of impurities (which ones will be discussed later) - C (mg/m3). The unit of measurement is the mass of impurities contained in 1 cubic meter of air, expressed in milligrams. The incidence rate will be characterized by the number of chronic asthma patients per 1000 residents of a given city - P (patients/thousand).
Let us note an important qualitative difference between the dependencies described in examples 1 and 2, on the one hand, and in example 3, on the other. In the first case, the relationship between the quantities is completely defined: the value of H uniquely determines the value of t (example 1), the value of t uniquely determines the value of P (example 2). But in the third example, the relationship between the value of air pollution and the level of morbidity is significantly more complex; With the same level of pollution in different months in the same city (or in different cities in the same month), the incidence rate may be different, since it is influenced by many other factors. We will postpone a more detailed discussion of this example until the next paragraph, but for now we will only note that in mathematical language the dependencies in examples 1 and 2 are functional, but in example 3 they are not.
Mathematical models
If the relationship between quantities can be represented in mathematical form, then we have a mathematical model.
A mathematical model is a set of quantitative characteristics of a certain object (process) and the connections between them, presented in the language of mathematics.
Mathematical models for the first two examples are well known. They reflect physical laws and are presented in the form of formulas:
In more complex problems, mathematical models are represented as equations or systems of equations. At the end of this chapter, we will consider an example of a mathematical model that is expressed by a system of inequalities.
In even more complex problems (example 3 is one of them), dependencies can also be represented in a mathematical form, but not a functional one, but a different one.
Tabular and graphical models
Let's look at examples of two other, non-formula, ways of presenting dependencies between quantities: tabular and graphical. Imagine that we decided to test the law of free fall of a body experimentally. We will organize the experiment as follows: we will throw a steel ball from a height of 6 meters, 9 meters, etc. (after 3 meters), measuring the height of the initial position of the ball and the time of fall. Based on the results of the experiment, we will create a table and draw a graph.
If each pair of values of H and t from this table is substituted into the above formula for the dependence of height on time, then the formula will turn into an equality (to within the measurement error). This means the model works well. (However, if you drop not a steel ball, but a large light ball, then equality will not be achieved, and if it is an inflatable ball, then the values of the left and right sides of the formula will differ very much. Why do you think?)
In this example, we looked at three ways to model the dependence of quantities: functional (formula), tabular and graphical. However, only a formula can be called a mathematical model of the process of a body falling to the ground. The formula is more universal; it allows you to determine the time of a body falling from any height, and not just for the experimental set of H values shown in Fig. 6.1. Having a formula, you can easily create a table and build a graph, but vice versa - it is very problematic.
In the same way, you can display the dependence of pressure on temperature in three ways. Both examples are related to known physical laws - the laws of nature. Knowledge of physical laws allows us to make accurate calculations; they form the basis of modern technology.
Information models that describe the development of systems over time have a special name: dynamic models. Example 1 shows just such a model. In physics, dynamic information models describe the movement of bodies, in biology - the development of organisms or animal populations, in chemistry - the course of chemical reactions, etc.
System of basic concepts
