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 students than the usual 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 capacity 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 an important topic that should not be missed in any case. 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!
Example
1.6 / 2 = 0.8; 4 / 5 = 0.8; 5.6 / 7 = 0.8 etc.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
Wikimedia Foundation. 2010 .
Solving problems from the problem book Vilenkin, Zhokhov, Chesnokov, Schwarzburd for grade 6 in mathematics on the topic:
§ 4. Relations and proportions:
22. Direct and inverse proportions
1 For 3.2 kg of goods they paid 115.2 rubles. How much should I pay for 1.5kg of this product?
DECISION
2 Two rectangles have the same area. The length of the first rectangle is 3.6 m and the width is 2.4 m. The length of the second is 4.8 m. Find its width.
DECISION
782 Determine whether the relationship between the following quantities is direct, inverse, or not proportional: the path traveled by a car at a constant speed and the time of its movement; the cost of goods purchased at one price, and its quantity; the area of the square and the length of its side; the mass of the steel bar and its volume; the number of workers performing some work with the same labor productivity, and the time of completion; the cost of the goods and its quantity, bought for a certain amount of money; the age of the person and the size of his shoes; the volume of the cube and the length of its edge; the perimeter of the square and the length of its side; a fraction and its denominator if the numerator does not change; fraction and its numerator if the denominator does not change.
DECISION
783 A steel ball with a volume of 6 cm3 has a mass of 46.8 g. What is the mass of a ball of the same steel if its volume is 2.5 cm3?
DECISION
784 5.1 kg of oil were obtained from 21 kg of cottonseed. How much oil will be obtained from 7 kg of cottonseed?
DECISION
785 For the construction of the stadium, 5 bulldozers cleared the site in 210 minutes. How long will it take 7 bulldozers to clear this site?
DECISION
786 It took 24 trucks with a carrying capacity of 7.5 tons to transport the cargo. How many trucks with a carrying capacity of 4.5 tons are needed to transport the same cargo?
DECISION
787 To determine the germination of seeds, peas were sown. Of the 200 peas sown, 170 sprouted. What percentage of the peas sprouted (germination)?
DECISION
788 Linden trees were planted on the street during Sunday Sunday to plant greenery in the city. 95% of all planted lindens were accepted. How many were planted if 57 linden trees were planted?
DECISION
789 There are 80 students in the ski section. Among them, 32 girls. What percentage of the participants in the section are girls and boys?
DECISION
790 The plant was supposed to smelt 980 tons of steel per month according to the plan. But the plan was fulfilled by 115%. How many tons of steel did the plant smelt?
DECISION
791 In 8 months, the worker completed 96% of the annual plan. What percentage of the annual plan will the worker fulfill in 12 months if he works with the same productivity?
DECISION
792 In three days, 16.5% of all beets were harvested. How many days will it take to harvest 60.5% of the beets if you work with the same productivity?
DECISION
793 B iron ore 7 parts of iron account for 3 parts of impurities. How many tons of impurities are in an ore that contains 73.5 tons of iron?
DECISION
794 To prepare borscht, for every 100 g of meat you need to take 60 g of beets. How many beets should be taken for 650 g of meat?
DECISION
796 Express as the sum of two fractions with a numerator of 1 each of the following fractions.
DECISION
797 From the numbers 3, 7, 9 and 21 make two correct proportions.
DECISION
798 Middle terms of proportion 6 and 10. What can be extreme terms? Give examples.
DECISION
799 At what value of x is the proportion correct.
DECISION
800 Find the ratio of 2 min to 10 s; 0.3 m2 to 0.1 dm2; 0.1 kg to 0.1 g; 4 hours to 1 day; 3 dm3 to 0.6 m3
DECISION
801 Where on the coordinate ray must the number c be located in order for the proportion to be correct.
DECISION
802 Cover the table with a sheet of paper. Open the first line for a few seconds and then, closing it, try to repeat or write down the three numbers of this line. If you correctly reproduced all the numbers, go to the second row of the table. If a mistake is made in any line, write several sets of the same number of two-digit numbers yourself and practice memorization. If you can reproduce at least five two-digit numbers without errors, you have a good memory.
DECISION
804 Is it possible to make the correct proportion of the following numbers.
DECISION
805 From the equality of products 3 · 24 = 8 · 9 make three correct proportions.
DECISION
806 The length of segment AB is 8 dm, and the length of segment CD is 2 cm. Find the ratio of the lengths of AB and CD. What part of AB is the length of CD?
DECISION
807 A voucher to a sanatorium costs 460 rubles. The trade union pays 70% of the ticket price. How much will a vacationer pay for a ticket?
DECISION
808 Find the value of the expression.
DECISION
809 1) When processing a part from a casting weighing 40 kg, 3.2 kg went to waste. What percentage is the mass of the part from the casting? 2) When sorting grain out of 1750 kg, 105 kg went to waste. What percentage of grain is left?
Example
1.6 / 2 = 0.8; 4 / 5 = 0.8; 5.6 / 7 = 0.8 etc.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
Wikimedia Foundation. 2010 .
- Newton's second law
- Coulomb barrier
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
PROPORTIONALITY- (from lat. proportionalis proportionate, proportional). Proportionality. Dictionary of 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, 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
Proportionality- Two mutually dependent quantities are called proportional if the ratio of their values \u200b\u200bremains unchanged .. Contents 1 Example 2 Proportionality coefficient ... Wikipedia
PROPORTIONALITY- 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
proportionality- 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
Dependency Types
Consider battery charging. As the first value, let's take the time it takes to charge. The second value is the time that it will work after charging. The longer the battery is charged, the longer it will last. The process will continue until the battery is fully charged.
The dependence of battery life on the time it is charged
Remark 1
This dependency is called straight:
As one value increases, the other also increases. As one value decreases, the other value also decreases.
Let's consider another example.
The more books the student reads, the fewer mistakes he will make in the dictation. Or the higher you climb the mountains, the lower the atmospheric pressure will be.
Remark 2
This dependency is called reverse:
As one value increases, the other decreases. As one value decreases, the other value increases.
Thus, in the case direct dependency both quantities change in the same way (both either increase or decrease), and in the case inverse relationship - opposite (one increases and the other decreases, or vice versa).
Determining dependencies between quantities
Example 1
The time it takes to visit a friend is $20$ minutes. With an increase in speed (of the first value) by $2$ times, we will find how the time (second value) that will be spent on the path to a friend will change.
Obviously, the time will decrease by $2$ times.
Remark 3
This dependency is called proportional:
How many times one value changes, how many times the second will change.
Example 2
For a $2 loaf of bread in a store, you have to pay 80 rubles. If you need to buy $4$ loaves of bread (the amount of bread increases $2$ times), how much more will you have to pay?
Obviously, the cost will also increase by $2$ times. We have an example of proportional dependence.
In both examples, proportional dependencies were considered. But in the example with loaves of bread, the values \u200b\u200bchange in one direction, therefore, the dependence is straight. And in the example with a trip to a friend, the relationship between speed and time is reverse. Thus, there is directly proportional relationship and inversely proportional relationship.
Direct proportionality
Consider $2$ proportional quantities: the number of loaves of bread and their cost. Let $2$ loaves of bread cost $80$ rubles. With an increase in the number of rolls by $4$ times ($8$ rolls), their total cost will be $320$ rubles.
The ratio of the number of rolls: $\frac(8)(2)=4$.
Roll cost ratio: $\frac(320)(80)=4$.
As you can see, these ratios are equal to each other:
$\frac(8)(2)=\frac(320)(80)$.
Definition 1
The equality of two relations is called proportion.
With a directly proportional relationship, a ratio is obtained when the change in the first and second values \u200b\u200bis the same:
$\frac(A_2)(A_1)=\frac(B_2)(B_1)$.
Definition 2
The two quantities are called directly proportional if, when changing (increasing or decreasing) one of them, the other value changes (increases or decreases accordingly) by the same amount.
Example 3
The car traveled $180$ km in $2$ hours. Find the time it takes for him to cover $2$ times the distance with the same speed.
Decision.
Time is directly proportional to distance:
$t=\frac(S)(v)$.
How many times the distance will increase, at a constant speed, the time will increase by the same amount:
$\frac(2S)(v)=2t$;
$\frac(3S)(v)=3t$.
The car traveled $180$ km - in the time of $2$ hour
The car travels $180 \cdot 2=360$ km - in the time of $x$ hours
The more distance the car travels, the more time it will take. Therefore, the relationship between the quantities is directly proportional.
Let's make a proportion:
$\frac(180)(360)=\frac(2)(x)$;
$x=\frac(360 \cdot 2)(180)$;
Answer: The car will need $4$ hours.
Inverse proportionality
Definition 3
Decision.
Time is inversely proportional to speed:
$t=\frac(S)(v)$.
How many times the speed increases, with the same path, the time decreases by the same amount:
$\frac(S)(2v)=\frac(t)(2)$;
$\frac(S)(3v)=\frac(t)(3)$.
Let's write the condition of the problem in the form of a table:
The car traveled $60$ km - in the time of $6$ hours
A car travels $120$ km - in a time of $x$ hours
The faster the car, the less time it will take. Therefore, the relationship between the quantities is inversely proportional.
Let's make a proportion.
Because proportionality is inverse, we turn the second ratio in proportion:
$\frac(60)(120)=\frac(x)(6)$;
$x=\frac(60 \cdot 6)(120)$;
Answer: The car will need $3$ hours.