How to prove direct and inverse proportionality. Practical application of direct and inverse proportionality

Example

Proportionality factor

The constant ratio of proportional quantities is called coefficient of proportionality. The proportionality coefficient shows how many units of one quantity fall on a unit of another.

Direct proportionality

Direct proportionality- functional dependence, in which some quantity depends on another quantity in such a way that their ratio remains constant. In other words, these variables change proportionately, in equal shares, that is, if the argument has changed twice in any direction, then the function also changes twice in the same direction.

Mathematically, direct proportionality is written as a formula:

f(x) = ax,a = const

Inverse proportionality

Inverse proportion- this is a functional dependence, in which an increase in the independent value (argument) causes a proportional decrease in the dependent value (function).

Mathematically, inverse proportionality is written as a formula:

Function properties:

Sources

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See what "Direct proportionality" is in other dictionaries:

direct proportionality- - [A.S. Goldberg. English Russian Energy Dictionary. 2006] Topics energy in general EN direct ratio … Technical Translator's Handbook

direct proportionality- tiesioginis proporcingumas statusas T sritis fizika atitikmenys: angl. direct proportionality vok. direkte Proportionalitat, f rus. direct proportionality, f pranc. proportionnalité directe, f … Fizikos terminų žodynas

- (from lat. proportionalis proportionate, proportional). Proportionality. Vocabulary foreign words included in the Russian language. Chudinov A.N., 1910. PROPORTIONALITY otlat. proportionalis, proportional. Proportionality. Explanation of 25000… … Dictionary of foreign words of the Russian language

PROPORTIONALITY, proportionality, pl. no, female (book). 1. distraction noun to proportional. Proportionality of parts. Body proportionality. 2. Such a relationship between quantities when they are proportional (see proportional ... Dictionary Ushakov

Two mutually dependent quantities are called proportional if the ratio of their values ​​\u200b\u200bremains unchanged .. Contents 1 Example 2 Proportionality coefficient ... Wikipedia

PROPORTIONALITY, and, wives. 1. see proportional. 2. In mathematics: such a relationship between quantities, when an increase in one of them entails a change in the other by the same amount. Direct p. (when cut with an increase in one value ... ... Explanatory dictionary of Ozhegov

AND; and. 1. to Proportional (1 digit); proportionality. P. parts. P. physique. P. representation in parliament. 2. Math. Dependence between proportionally changing quantities. Proportionality factor. Direct p. (In which with ... ... encyclopedic Dictionary

I. Directly proportional quantities.

Let the value y depends on the size X. If with an increase X several times the size at increases by the same factor, then such values X and at are called directly proportional.

Examples.

1 . The quantity of the purchased goods and the cost of the purchase (at a fixed price of one unit of goods - 1 piece or 1 kg, etc.) How many times more goods were bought, so many times more and paid.

2 . The distance traveled and the time spent on it (at constant speed). How many times longer the path, how many times more time we will spend on it.

3 . The volume of a body and its mass. ( If one watermelon is 2 times larger than the other, then its mass will be 2 times larger)

II. The property of direct proportionality of quantities.

If two quantities are directly proportional, then the ratio of two arbitrary values ​​of the first quantity is equal to the ratio of the two corresponding values ​​of the second quantity.

Task 1. For raspberry jam 12 kg raspberries and 8 kg Sahara. How much sugar will be required if taken 9 kg raspberries?

Decision.

We argue like this: let it be necessary x kg sugar on 9 kg raspberries. The mass of raspberries and the mass of sugar are directly proportional: how many times less raspberries, the same amount of sugar is needed. Therefore, the ratio of taken (by weight) raspberries ( 12:9 ) will be equal to the ratio of sugar taken ( 8:x). We get the proportion:

12: 9=8: X;

x=9 · 8: 12;

x=6. Answer: on 9 kg raspberries to take 6 kg Sahara.

The solution of the problem could have been done like this:

Let on 9 kg raspberries to take x kg Sahara.

(The arrows in the figure are directed in one direction, and it does not matter up or down. Meaning: how many times the number 12 more number 9 , the same number 8 more number X, i.e., there is a direct dependence here).

Answer: on 9 kg raspberries to take 6 kg Sahara.

Task 2. car for 3 hours traveled distance 264 km. How long will it take him 440 km if it travels at the same speed?

Decision.

Let for x hours the car will cover the distance 440 km.

Answer: the car will pass 440 km in 5 hours.

You can talk endlessly about the advantages of learning with the help of video lessons. First, they express thoughts clearly and understandably, consistently and structured. Secondly, they take a certain fixed time, are not, often stretched and tedious. Thirdly, they are more exciting for schoolchildren than regular lessons to which they are accustomed. You can view them in a relaxed atmosphere.

In many tasks from the mathematics course, students in grade 6 will encounter direct and inverse proportionality. Before starting the study of this topic, it is worth remembering what proportions are and what basic property they have.

The topic “Proportions” is devoted to the previous video lesson. This one is a logical continuation. It is worth noting that the topic is quite important and often encountered. It should be properly understood once and for all.

To show the importance of the topic, the video tutorial starts with a task. The condition appears on the screen and is announced by the announcer. The data recording is given in the form of a diagram so that the student viewing the video recording can understand it as best as possible. It would be better if for the first time he adheres to this form of recording.

The unknown, as is customary in most cases, is denoted by the Latin letter x. To find it, you must first multiply the values ​​crosswise. Thus, the equality of the two ratios will be obtained. This suggests that it has to do with proportions and it is worth remembering their main property. Please note that all values ​​are given in the same unit of measure. Otherwise, it was necessary to bring them to the same dimension.

After viewing the solution method in the video, there should not be any difficulties in such tasks. The announcer comments on each move, explains all the actions, recalls the studied material that is used.

Immediately after watching the first part of the video lesson “Direct and inverse proportional relationships”, you can offer the student to solve the same problem without the help of prompts. After that, an alternative task can be proposed.

Depending on the mental abilities of the student, you can gradually increase the complexity of subsequent tasks.

After the first considered problem, the definition of directly proportional quantities is given. The definition is read out by the announcer. The main concept is highlighted in red.

Next, another problem is demonstrated, on the basis of which the inverse proportional relationship is explained. It is best for the student to write these concepts in a notebook. If necessary before control work, the student can easily find all the rules and definitions and reread.

After watching this video, a 6th grader will understand how to use proportions in certain tasks. This is enough important topic not to be missed under any circumstances. If the student is not adapted to perceive the material presented by the teacher during the lesson among other students, then such learning resources will be a great salvation!