Quantities are the quantitative values of objects, lengths of segments, time, angles, etc.
Definition. Value - the result of a measurement, represented by a number and the name of the unit of measurement.
For example: 1 km; 5 hours 60 km/h; 15 kg; 180°.
Quantities may be independent or dependent on one another. The relationship of quantities can be rigidly established (for example, 1 dm \u003d 10 cm) or may reflect the relationship between quantities, expressed by a formula for determining a specific numerical value (for example, the path depends on the speed and duration of movement; the area of the square - on its length sides, etc.).
The basis of the metric system of measures of length - the meter - was introduced in Russia at the beginning of the 19th century, and before that, the following were used to measure lengths: arshin (= 71 cm), verst (= 1067 m), oblique sazhen (= 2 m 13 cm), flywheel fathom (= 1 m 76 cm), simple fathom (= 1 m 52 cm), quarter (= 18 cm), cubit (approximately from 35 cm to 46 cm), span (from 18 cm to 23 cm).
As you can see, there were many quantities to measure length. With the introduction of the metric system of measures, the dependence of the length values is rigidly fixed:
- 1 km = 1,000 m; 1 m = 100 cm;
- 1 dm = 10 cm; 1 cm = 10 mm.
In the metric system of measures, units of measurement for time, length, mass, volume, area and speed are defined.
Between two or more quantities or systems of measures, it is also possible to establish a relationship, it is fixed in the formulas, and the formulas are derived empirically.
Definition. Two mutually dependent quantities are called proportional if the ratio of their values remains unchanged.
The constant ratio of two quantities is called the coefficient of proportionality. Proportionality factor shows how many units of one quantity are per unit of another quantity. If the coefficients are equal. That relationship is equal.
Distance is the product of speed and time of movement: from here the basic formula for movement was derived:
where S- path; V- speed; t- time.
The basic formula of movement is the dependence of distance on speed and time of movement. This dependency is called spicy proportional.
Definition. Two variables are directly proportional if, with an increase (or decrease) several times in one value, the other value increases (or decreases) by the same amount; those. the ratio of the corresponding values of such quantities is a constant value.
At a constant distance, speed and time are related by another relationship, which is called inversely proportional.
Rule. Two variables are inversely proportional if, with an increase (or decrease) in one value several times, the other value decreases (or increases) by the same amount; those. the product of the corresponding values of such quantities is a constant value.
Two more relations can be deduced from the motion formula, expressing direct and inverse dependences of the quantities included in them:
t=S:V- travel time in direct ratio path traveled and inversely speed of movement (for the same segments of the path, the greater the speed, the less time is required to overcome the distance).
V=S:t- movement speed directly proportional path traveled and inversely proportional travel time (for the same segments of the path, the more
time an object moves, the less speed is required to overcome distances).
All three motion formulas are equivalent and are used to solve problems.
Development of a math lesson in grade 6The topic of the lesson is "Dependence between quantities."
Lesson Objectives:
1. Give the concept of dependence between quantities, find out how to set them.
2. To develop the ability of students to analyze and synthesize educational material.
3. Bring up a creative attitude to educational work.
4. Present educational material through the emotional-experiential sphere of the student.
And now we will describe the technology of construction by the teacher of the methodology of the lesson according to the technology of the activity method.
1. The stage of self-determination of the norm N
At this stage, the topic and the learning goal of the lesson are determined: “In the lesson we will consider the relationship between different quantities”, that is, the operation is declared without specifying the conditions for its application.
2. The stage of updating knowledge and fixing difficulties in activities.
At this stage, the teacher offers a list of tasks, the implementation of which involves the fulfillment of a previously known norm.
How to find:
Area of a rectangle?
Perimeter of a rectangle?
The volume of a rectangular parallelepiped?
Downstream speed?
Speed against the current?
The last question at the stage of updating knowledge should be a question that fixes difficulties in the activities of students, that is, previously studied knowledge is not enough, a learning problem arises. In this case, this is the question: “What are these rules and the corresponding formulas for?”.
3. The stage of setting a learning task.
The teacher poses a problem for students: How to measure the area of a rectangular plot if we do not know the formulaS=av? You can divide the site into rectangles of 1 square. meter and count their number. Is it convenient?
Students answer that it is possible, but inconvenient. This means that formulas are needed to calculate quantities that are difficult to measure.
The teacher poses an even more compelling problem: how to measure the distance from the Earth to the Sun? So, there is a crisis of the previously known normN.
4. The stage of building a project to get out of difficulty.
Scientists have established that the distance from the Earth to the Sun is 150 million km. And how did they know about it? Together with the children, the formula for calculating the distance from the Earth to the Sun is found outs= ct, where c=300000km,t\u003d 8 min, the time it takes for light to reach the Earth. Calculations show thats=2400000 km. Why did we get a discrepancy with a known fact?
Conclusion: The formula can be applied only if the units of measurement of the quantities included in it are consistent with each other.
At this stage, it is appropriate to influence the emotional and emotional sphere of the student with the help of a small educational conversation. “Light travels from the Earth to the Sun within 8 minutes, which means that we see the Sun as it was 8 minutes ago. There are stars whose light comes to us for millions of years: the star may have already gone out, but the light from it is still coming. There are people in the same way: a person is no longer with us, and his warmth, light warms us all our lives. Such a person was the national poet of Bashkortostan Mustai Karim, whose memorial day we celebrate today. His spiritual energy, the warmth of his heart will serve us as a moral guide for many years.”
At this stage of the lesson, students are offered various ways to set dependencies between quantities: tabular, graphical, and using a formula.
Children at this stage are included in the situation of choosing a method for solving a learning problem: they compare different ways of specifying dependencies between quantities. The results of the comparison are recorded on the support-nodal matrix.
1 2Ways of setting Formula graph table
1-universality, 2-accuracy, 3-visibility;
(Symbols "D" - yes, "N" - no)
Based on the analysis of the support-nodal matrix, students conclude that the best way is to set the relationship between quantities using a formula, because it has the property of universality: you can get a dependency table from the formula and build a dependency graph between quantities.
5. The stage of primary consolidation in external speech.
Problem No. 90 is being analyzed
According to one formula for the dependence of the width of a rectangle on its length with a constant area:b=12/а to make a table of this dependence and build its graph.
1 ,5
1,5
Rectangle length versus width graph
So, we have linked 3 ways of specifying dependencies between quantities:
With the formula,
Graphic,
Tabular.
6. The stage of independent work with self-testing according to the standard.
Students independently solve tasks for a new way of acting, perform self-examination according to the standard and evaluate their own results. A situation of success is created, the emotional-experiential sphere of the student is again involved. At one stage, students are offered tasks No. 133, No. 140. To implement the minimax principle of the activity learning technology, students are offered tasks of two levels: M, A and B.
Level M: #133, A: #140. Level B: No. 145
7. Incorporating new knowledge into knowledge.
At this stage, students are convinced that the newly acquired knowledge is of value for further learning. Performing exercise No. 139, they establish a relationship between
VolumeVcube and its edge a;
areaSright triangle and legs a andb
diameterDand radiusRthis circle;
The length of the side a of the rectangle, its perimeter P and areaS;
Scube and its edge a
Full surface areaScuboid and its dimensions a,band s.
8. Reflection of activity (the result of the lesson)
Students perform self-assessment of their own activities (what new things they learned, what method they used, the success of the steps taken). There is a fixation of the success of the activity and a conclusion about the next steps. The students who completed the tasks of level A and B are identified.
Note.
The lesson was conducted according to the textbook by G.V. Dorofeev, L.G. Peterson. Mathematics, textbook for grade 6. Part 2. Juventa. 2011
Subject: mathematics
Class: 4
Lesson topic: Relationships between speed, distance traveled and time
movement.
Purpose: to identify and justify the relationship between the quantities: speed, time,
distance;
Tasks: to promote the development of non-standard thinking, the ability to draw conclusions,
reason; contribute to the education of cognitive activity.
Equipment: individual cards in different colors, evaluation criteria,
reflection card, circles of two colors.
During the classes.
1. Organizing moment.
Card in two colors: yellow and blue. Show your mood with a card
at the beginning and end of the lesson.
Filling out the card at the beginning of the lesson (Appendix 1.)
No. Approval
End of the lesson
Lesson start
Yes
Not
Don't know Yes
Not no
I know
1. I know all the formulas
movement tasks
2. I understand the decision
movement tasks
3. I can decide for myself
tasks
4. I can compose
schemes for tasks
traffic
5. I know what mistakes
admit in the decision
movement tasks
2. Repetition.
How to find speed? Time? Distance?
What are the units of measure for speed, distance, time.
3. Message of the topic of the lesson.
What will we learn in class?
4. Work in a group.
Connect motion objects (Appendix 2)
Pedestrian 70km/h
Skier 5km/h
Car 10km/h
Jet plane 12km/h
Train 50km/h
Snail 900km/h
Horse 90 km/h
Checking work.
5. Mathematical puzzle (independent work)
How much is the speed of the cyclist less than the speed of the train?
How many km is the skier's speed faster than the walking speed?
How many times the speed of a car is less than the speed of a jet plane?
Find the combined speed of the fastest moving vehicle and the fastest
slow.
Find the combined speed of the cyclist and skier train.
6. Self-checking of works according to the criteria.
7. Physical Minute.
The red color of the square is standing
Green - let's go
Yellow - clap your hands 1 time
8. Work in a group. (Card yellow) (Jegso method)
A task.
Two women argued that a stupa or a pomelo was faster? The same
a distance of 228 km was covered by a babayaga in a mortar in 4 hours, and a babayaga on a broom in 3 hours. What
more speed stupa or pomelo?
9. Work in a pair of "Experiment".
Come up with a movement problem using the following values: 18km/h, 4h, 24km, 3h.
Checking work.
10. Test.
1. Write down the formula for finding the speed.
2. Write down the formula for finding time.
3. How to find the distance? Write down the formula.
4. Write down 8 km/min in km/h
5. Find the time it takes for a pedestrian to walk 42 km, moving at a speed of 5 km/h.
6. How far will a pedestrian travel, moving at a speed of 5 km / h for 6 hours?
11. The result of the lesson.
Fill in the table with what results we came to the end of the lesson.
Show a card that matches your mood.
Lesson start
Yes
Not
Attachment 1.
End of the lesson
Don't know Yes
No. Approval
1. I know all the formulas
movement tasks
2. I understand the decision
movement tasks
3. I can decide for myself
tasks
4. I can compose
schemes for tasks
traffic
5. I know what mistakes
admit in the decision
movement tasks
Connect motion objects.
Pedestrian 70km/h
Skier 5km/h
Car 10km/h
Jet plane 12km/h
Train 50km/h
Snail 900km/h
Horse 90 km/h
Not no
I know
Appendix 2
The dependence of one random variable on the values that another random variable (physical characteristic) takes on is usually called regression in statistics. If this dependence is given an analytical form, then this form of presentation is represented by a regression equation.
The procedure for searching for the alleged relationship between different numerical populations usually includes the following steps:
establishing the significance of the relationship between them;
the possibility of representing this dependence in the form of a mathematical expression (regression equation).
The first step in this statistical analysis concerns the identification of the so-called correlation, or correlation dependence. Correlation is considered as a sign indicating the relationship of a number of numerical sequences. In other words, correlation characterizes the strength of the relationship in the data. If it concerns the relationship of two numerical arrays xi and yi, then such a correlation is called paired.
When searching for a correlation, a probable connection of one measured value x (for some limited range of its change, for example, from x1 to xn) with another measured value y (also changing in some interval y1 ... yn) is usually revealed. In this case, we will be dealing with two numerical sequences, between which it is necessary to establish the presence of a statistical (correlation) relationship. At this stage, the task is not yet set to determine whether one of these random variables is a function, and the other is an argument. Finding a quantitative relationship between them in the form of a specific analytical expression y = f(x) is the task of another analysis, regression.
Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, correlation analysis infers the strength of the relationship between data pairs x and y, while regression analysis is used to predict one variable (y) based on another (x). In other words, in this case, they try to identify a causal relationship between the analyzed populations.
Strictly speaking, it is customary to distinguish between two types of connection between numerical sets - ϶ᴛᴏ can be a functional dependence or a statistical (random) one. In the presence of a functional connection, each value of the influencing factor (argument) corresponds to a strictly defined value of another indicator (function), ᴛ.ᴇ. the change in the effective attribute is entirely due to the action of the factor attribute.
Analytically, the functional dependence is presented in the following form: y = f(x).
In the case of a statistical relationship, the value of one factor corresponds to some approximate value of the parameter under study, its exact value is unpredictable, unpredictable, and therefore the resulting indicators turn out to be random variables. This means that the change in the effective attribute y is due to the influence of the factor attribute x only partially, because the influence of other factors is also possible, the contribution of which is indicated as є: y = f(x) + є.
By their nature, correlations are ϶ᴛᴏ correlative connections. An example of a correlation between indicators of commercial activity is, for example, the dependence of the amounts of distribution costs on the volume of trade. In this regard, in addition to the factor sign x (volume of goods turnover), the effective sign y (the sum of distribution costs) is also influenced by other factors, including unaccounted ones, that generate the contribution є.
For a quantitative assessment of the existence of a connection between the studied sets of random variables, a special statistical indicator is used - the correlation coefficient r.
If it is assumed that this relationship can be described by a linear equation of the type y=a+bx (where a and b are constants), then it is customary to talk about the existence of a linear correlation.
The coefficient r is a dimensionless quantity, it can vary from 0 to ±1. The closer the value of the coefficient is to one (no matter with what sign), the more confident it can be argued that there is a linear relationship between the two sets of variables under consideration. In other words, the value of one of these random variables (y) essentially depends on what value the other one (x) takes.
If it turns out that r = 1 (or -1), then there is a classic case of purely functional dependence (ᴛ.ᴇ. an ideal relationship is realized).
When analyzing a two-dimensional scatterplot, various relationships can be found. The simplest option is a linear relationship, which means that points are placed randomly along a straight line. The diagram indicates no relationship if the points are randomly located and no slope (neither up nor down) can be detected when moving from left to right.
If the points on it are grouped along a curved line, then the scatterplot is characterized by a non-linear relationship. Such situations are quite possible.
Regression Analysis
Processing the results of the experiment by the method
When studying the processes of functioning of complex systems, one has to deal with a number of simultaneously acting random variables. To understand the mechanism of phenomena, cause-and-effect relationships between the elements of the system, etc., we are trying to establish the relationship of these quantities based on the observations received.
In mathematical analysis, the dependence, for example, between two quantities is expressed by the concept of a function
where each value of one variable corresponds to only one value of the other. This dependence is called functional.
The situation with the concept of dependence of random variables is much more complicated. As a rule, between random variables (random factors) that determine the process of functioning of complex systems, there is usually such a relationship in which, with a change in one variable, the distribution of another changes. Such a connection is called stochastic, or probabilistic. In this case, the magnitude of the change in the random factor Y, corresponding to the change in the value X, can be broken down into two components. The first is related to addiction. Y from X, and the second with the influence of "own" random components Y and X. If the first component is missing, then the random variables Y and X are independent. If the second component is missing, then Y and X depend functionally. In the presence of both components, the ratio between them determines the strength or tightness of the relationship between random variables Y and X.
There are various indicators that characterize certain aspects of the stochastic relationship. So, a linear relationship between random variables X and Y determines the correlation coefficient.
where are the mathematical expectations of random variables X and Y.
– standard deviations of random variables X and Y.
The linear probabilistic dependence of random variables lies in the fact that as one random variable increases, the other tends to increase (or decrease) according to a linear law. If random variables X and Y are connected by a strict linear functional dependence, for example,
y=b 0 +b 1 x 1,
then the correlation coefficient will be equal to ; where the sign corresponds to the sign of the coefficient b 1.If the values X and Y are connected by an arbitrary stochastic dependence, then the correlation coefficient will vary within
It should be emphasized that for independent random variables, the correlation coefficient zero. However, the correlation coefficient as an indicator of the dependence between random variables has serious drawbacks. First, from the equality r= 0 does not imply independence of random variables X and Y(with the exception of random variables subject to the normal distribution law, for which r= 0 means at the same time the absence of any dependence). Secondly, the extreme values are also not very useful, since they do not correspond to any functional dependence, but only to a strictly linear one.
Full dependency description Y from X, and, moreover, expressed in exact functional relationships, can be obtained by knowing the conditional distribution function .
It should be noted that in this case one of the observed variables is considered nonrandom. Fixing simultaneously the values of two random variables X and Y, when comparing their values, we can attribute all errors only to the value Y. Thus, the observation error will be the sum of its own random error of the quantity Y and from the matching error arising from the fact that with the value Y not quite the same value is matched X which actually took place.
However, finding the conditional distribution function, as a rule, turns out to be a very difficult task. The easiest way to investigate the relationship between X and Y with a normal distribution Y, since it is completely determined by the mathematical expectation and variance. In this case, to describe the dependence Y from X you do not need to build a conditional distribution function, but just indicate how, when changing the parameter X the mathematical expectation and variance of the value change Y.
Thus, we come to the need to find only two functions:
(3.2)
Conditional variance dependence D from parameter X is called skhodastichesky dependencies. It characterizes the change in the accuracy of the observation technique with a change in the parameter and is used quite rarely.
Dependence of the conditional mathematical expectation M from X is called regression, it gives the true dependence of the quantities X and At, devoid of all random layers. Therefore, the ideal goal of any study of dependent variables is to find a regression equation, and the variance is used only to assess the accuracy of the result.
