How to find out direct or inverse proportionality. Direct proportionality
Example
1.6 / 2 = 0.8; 4 / 5 = 0.8; 5.6 / 7 = 0.8, etc.Proportionality factor
A constant relationship of proportional quantities is called proportionality factor. The proportionality coefficient shows how many units of one quantity are per unit of another.
Direct proportionality
Direct proportionality- functional dependence, in which a certain quantity depends on another quantity in such a way that their ratio remains constant. In other words, these variables change proportionally, in equal shares, that is, if the argument changes 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 proportionality- 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.
We can talk endlessly about the advantages of learning using video lessons. Firstly, they present their thoughts clearly and understandably, consistently and in a structured manner. Secondly, they take a certain fixed time and are not often drawn out and tedious. Thirdly, they are more exciting for students than the regular lessons they are used to. You can view them in a calm atmosphere.
In many problems from the mathematics course, 6th grade students will be faced with direct and inverse proportional relationships. Before you start studying this topic, it is worth remembering what proportions are and what basic properties they have.
The previous video lesson is devoted to the topic “Proportions”. This one is a logical continuation. It is worth noting that the topic is quite important and frequently encountered. It is worth understanding properly once and for all.
To show the importance of the topic, the video lesson begins with a task. The condition appears on the screen and is announced by the announcer. The data recording is given in the form of some kind of diagram so that the student watching the video recording can understand as best as possible. It would be better if at first 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 indicated in the same unit of measurement. Otherwise, it was necessary to reduce them to one dimension.
After watching the solution method in the video, you should not have any difficulties with such problems. The announcer comments on each move, explains all the actions, and recalls the studied material that is used.
Immediately after watching the first part of the video lesson “Direct and inverse proportional dependencies”, you can ask the student to solve the same problem without the help of hints. Afterwards, you can offer an alternative task.
Depending on the student’s mental abilities, the difficulty of subsequent tasks can be gradually increased.
After the first problem considered, 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 down these concepts in a notebook. If necessary, before tests, the student can easily find all the rules and definitions and re-read.
After watching this video, a 6th grader will understand how to use proportions in certain tasks. This is a fairly important topic that should not be missed under any circumstances. If a student is not able to perceive the material presented by the teacher during a lesson among other students, then such educational resources will be a great salvation!
Completed by: Chepkasov Rodion
6th grade student
MBOU "Secondary School No. 53"
Barnaul
Head: Bulykina O.G.
mathematic teacher
MBOU "Secondary School No. 53"
Barnaul
Introduction. 1
Relationships and proportions. 3
Direct and inverse proportional relationships. 4
Application of direct and inverse proportional 6
dependencies when solving various problems.
Conclusion. eleven
Literature. 12
Introduction.
The word proportion comes from the Latin word proportion, which generally means proportionality, alignment of parts (a certain ratio of parts to each other). In ancient times, the doctrine of proportions was held in high esteem by the Pythagoreans. With proportions they associated thoughts about order and beauty in nature, about consonant chords in music and harmony in the universe. They called some types of proportions musical or harmonic.
Even in ancient times, man discovered that all phenomena in nature are connected with each other, that everything is in continuous movement, change, and, when expressed in numbers, reveals amazing patterns.
The Pythagoreans and their followers sought everything in the world numeric expression. They discovered; that mathematical proportions underlie music (the ratio of the length of the string to the pitch, the relationship between intervals, the ratio of sounds in chords that give a harmonic sound). The Pythagoreans tried to mathematically substantiate the idea of the unity of the world and argued that the basis of the universe was symmetrical geometric shapes. The Pythagoreans sought a mathematical basis for beauty.
Following the Pythagoreans, the medieval scientist Augustine called beauty “numerical equality.” The scholastic philosopher Bonaventure wrote: “There is no beauty and pleasure without proportionality, and proportionality exists primarily in numbers. It is necessary that everything be countable.” Leonardo da Vinci wrote about the use of proportion in art in his treatise on painting: “The painter embodies in the form of proportion the same patterns hidden in nature that the scientist knows in the form of the numerical law.”
Proportions were used to solve various problems in both antiquity and the Middle Ages. Certain types of problems are now easily and quickly solved using proportions. Proportions and proportionality were and are used not only in mathematics, but also in architecture and art. Proportion in architecture and art means maintaining certain relationships between sizes different parts building, figure, sculpture or other work of art. Proportionality in such cases is a condition for correct and beautiful construction and depiction
In my work, I tried to consider the use of direct and inverse proportional relationships in various areas surrounding life, trace the connection with academic subjects through tasks.
Relationships and proportions.
The quotient of two numbers is called attitude these numbers.
Attitude shows, how many times the first number is greater than the second or what part the first number is of the second.
Task.
2.4 tons of pears and 3.6 tons of apples were brought to the store. What proportion of the fruits brought are pears?
Solution . Let's find how much fruit they brought: 2.4+3.6=6(t). To find what part of the brought fruits are pears, we make the ratio 2.4:6=. The answer can also be written in the form decimal or as a percentage: = 0.4 = 40%.
Mutually inverse called numbers, whose products are equal to 1. Therefore the relationship is called the inverse of the relationship.
Consider two equal ratios: 4.5:3 and 6:4. Let's put an equal sign between them and get the proportion: 4.5:3=6:4.
Proportion is the equality of two relations: a : b =c :d or = 
, where a and d are extreme members proportions, c and b – average members(all terms of the proportion are different from zero).
Basic property of proportion:
in the correct proportion, the product of the extreme terms is equal to the product of the middle terms.
Applying the commutative property of multiplication, we find that in the correct proportion the extreme terms or middle terms can be interchanged. The resulting proportions will also be correct.
Using the basic property of proportion, you can find its unknown term if all other terms are known.
To find the unknown extreme term of the proportion, you need to multiply the average terms and divide by the known extreme term. x : b = c : d , x = 
To find the unknown average member proportions, you need to multiply the extreme terms and divide by the known middle term. a : b =x : d , x = 
.
Direct and inverse proportional relationships.
The values of two different quantities can be mutually dependent on each other. So, the area of a square depends on the length of its side, and vice versa - the length of the side of a square depends on its area.
Two quantities are said to be proportional if, with increasing
(decrease) one of them several times, the other increases (decreases) the same number of times.
If two quantities are directly proportional, then the ratios of the corresponding values of these quantities are equal.
Example direct proportional dependence .
At a gas station 2 liters of gasoline weigh 1.6 kg. How much will they weigh 5 liters of gasoline?
Solution:
The weight of kerosene is proportional to its volume.
2l - 1.6 kg
5l - x kg
2:5=1.6:x,
x=5*1.6 x=4
Answer: 4 kg.
Here the weight to volume ratio remains unchanged.
Two quantities are called inversely proportional if, when one of them increases (decreases) several times, the other decreases (increases) by the same amount.
If quantities are inversely proportional, then the ratio of the values of one quantity is equal to the inverse ratio of the corresponding values of another quantity.
P exampleinversely proportional relationship.
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 rectangle is 4.8 m. Find the width of the second rectangle.
Solution:
1 rectangle 3.6 m 2.4 m
2 rectangle 4.8 m x m
3.6 m x m
4.8 m 2.4 m
x = 3.6*2.4 = 1.8 m
Answer: 1.8 m.
As we can see, the tasks for proportional quantities can be solved using proportions.
Not every two quantities are directly proportional or inversely proportional. For example, a child’s height increases as his age increases, but these values are not proportional, since when the age doubles, the child’s height does not double.
Practical use direct and inverse proportional dependence.
Task No. 1
The school library has 210 mathematics textbooks, which is 15% of the entire library collection. How many books are there in the library collection?
Solution:
Total textbooks - ? - 100%
Mathematicians - 210 -15%
15% 210 academic.
X = 100* 210 = 1400 textbooks
100% x account. 15
Answer: 1400 textbooks.
Problem No. 2
A cyclist travels 75 km in 3 hours. How long will it take a cyclist to travel 125 km at the same speed?
Solution:
3 h – 75 km
H – 125 km
Time and distance are directly proportional quantities, therefore
3: x = 75: 125,
x= 
,
x=5.
Answer: in 5 hours.
Problem No. 3
8 identical pipes fill a pool in 25 minutes. How many minutes will it take to fill a pool with 10 such pipes?
Solution:
8 pipes – 25 minutes
10 pipes - ? minutes
The number of pipes is inversely proportional to time, so
8:10 = x:25,
x = 
x = 20
Answer: in 20 minutes.
Problem No. 4
A team of 8 workers completes the task in 15 days. How many workers can complete the task in 10 days while working at the same productivity?
Solution:
8 working days – 15 days
Workers - 10 days
The number of workers is inversely proportional to the number of days, so
x: 8 = 15: 10,
x= 
,
x=12.
Answer: 12 workers.
Problem No. 5
From 5.6 kg of tomatoes, 2 liters of sauce are obtained. How many liters of sauce can be obtained from 54 kg of tomatoes?
Solution:
5.6 kg – 2 l
54 kg - ? l
The number of kilograms of tomatoes is directly proportional to the amount of sauce obtained, therefore
5.6:54 = 2:x,
x = 
,
x = 19.
Answer: 19 l.
Problem No. 6
To heat the school building, coal was stored for 180 days at the consumption rate
0.6 tons of coal per day. How many days will this supply last if 0.5 tons are spent daily?
Solution:
Number of days
Consumption rate
The number of days is inversely proportional to the rate of coal consumption, therefore
180: x = 0.5: 0.6,
x = 180*0.6:0.5,
x = 216.
Answer: 216 days.
Problem No. 7
In iron ore, for every 7 parts iron there are 3 parts impurities. How many tons of impurities are in the ore that contains 73.5 tons of iron?
Solution:
Number of parts
Weight
Iron
73,5
Impurities
The number of parts is directly proportional to the mass, therefore
7: 73.5 = 3: x.
x = 73.5 * 3:7,
x = 31.5.
Answer: 31.5 t
Problem No. 8
The car traveled 500 km, using 35 liters of gasoline. How many liters of gasoline will be needed to travel 420 km?
Solution:
Distance, km
Gasoline, l
The distance is directly proportional to gasoline consumption, so
500:35 = 420:x,
x = 35*420:500,
x = 29.4.
Answer: 29.4 l
Problem No. 9
In 2 hours we caught 12 crucian carp. How many crucian carp will be caught in 3 hours?
Solution:
The number of crucian carp does not depend on time. These quantities are neither directly proportional nor inversely proportional.
Answer: There is no answer.
Problem No. 10
A mining enterprise needs to purchase 5 new machines for a certain amount of money at a price of 12 thousand rubles per one. How many of these machines can an enterprise buy if the price for one machine becomes 15 thousand rubles?
Solution:
Number of cars, pcs.
Price, thousand rubles
The number of cars is inversely proportional to the cost, so
5: x = 15: 12,
x=5*12:15,
x=4.
Answer: 4 cars.
Problem No. 11
In the city N on square P there is a store whose owner is so strict that for lateness he deducts 70 rubles from the salary for 1 lateness per day. Two girls Yulia and Natasha work in one department. Their wage depends on the number of working days. Yulia received 4,100 rubles in 20 days, and Natasha should have received more in 21 days, but she was late for 3 days in a row. How many rubles will Natasha receive?
Solution:
Work days
Salary, rub.
Julia
4100
Natasha
Salary is directly proportional to the number of working days, therefore
20:21 = 4100:x,
x=4305.
4305 rub. Natasha should have received it.
4305 – 3 * 70 = 4095 (rub.)
Answer: Natasha will receive 4095 rubles.
Problem No. 12
The distance between two cities on the map is 6 cm. Find the distance between these cities on the ground if the map scale is 1: 250000.
Solution:
Let us denote the distance between cities on the ground by x (in centimeters) and find the ratio of the length of the segment on the map to the distance on the ground, which will be equal to the map scale: 6: x = 1: 250000,
x = 6*250000,
x = 1500000.
1500000 cm = 15 km
Answer: 15 km.
Problem No. 13
4000 g of solution contains 80 g of salt. What is the concentration of salt in this solution?
Solution:
Weight, g
Concentration, %
Solution
4000
Salt
4000: 80 = 100: x,
x = 
,
x = 2.
Answer: The salt concentration is 2%.
Problem No. 14
The bank gives a loan at 10% per annum. You received a loan of 50,000 rubles. How much should you return to the bank in a year?
Solution:
50,000 rub.
100%
x rub.
50000: x = 100: 10,
x= 50000*10:100,
x=5000.
5000 rub. is 10%.
50,000 + 5000=55,000 (rub.)
Answer: in a year the bank will get 55,000 rubles back.
Conclusion.
As we can see from the examples given, direct and inverse proportional relationships are applicable in various areas of life:
Economics,
Trade,
In production and industry,
Cooking,
Construction and architecture.
Sports,
Animal husbandry,
Topographies,
Physicists,
Chemistry, etc.
In the Russian language there are also proverbs and sayings that establish direct and inverse relationship:
As it comes back, so will it respond.
The higher the stump, the higher the shadow.
The more people, the less oxygen.
And it’s ready, but stupid.
Mathematics is one of the ancient sciences, it arose on the basis of the needs and wants of humanity. Having gone through the history of formation since Ancient Greece, it still remains relevant and necessary in Everyday life any person. The concept of direct and inverse proportionality has been known since ancient times, since it was the laws of proportion that motivated architects during any construction or creation of any sculpture.
Knowledge about proportions is widely used in all spheres of human life and activity - one cannot do without it when painting (landscapes, still lifes, portraits, etc.), it is also widespread among architects and engineers - in general, it is difficult to imagine creating anything something without using knowledge about proportions and their relationships.
Literature.
Mathematics-6, N.Ya. Vilenkin et al.
Algebra -7, G.V. Dorofeev and others.
Mathematics-9, GIA-9, edited by F.F. Lysenko, S.Yu. Kulabukhova
Mathematics-6, didactic materials, P.V. Chulkov, A.B. Uedinov
Problems in mathematics for grades 4-5, I.V. Baranova et al., M. "Prosveshchenie" 1988
Collection of problems and examples in mathematics grades 5-6, N.A. Tereshin,
T.N. Tereshina, M. “Aquarium” 1997
I. Directly proportional quantities.
Let the value y depends on the size X. If when increasing X several times the size at increases by the same amount, then such values X And at are called directly proportional.
Examples.
1 . The quantity of goods purchased and the purchase price (with a fixed price for one unit of goods - 1 piece or 1 kg, etc.) How many times more goods were bought, the more times more they paid.
2 . The distance traveled and the time spent on it (with constant speed).How many times longer is the path, how many times more time will it take to complete it.
3 . The volume of a body and its mass. ( If one watermelon is 2 times larger than another, then its mass will be 2 times larger)
II. Property of direct proportionality of quantities.
If two quantities are directly proportional, then the ratio of two arbitrarily taken values of the first quantity is equal to the ratio of two corresponding values of the second quantity.
Task 1. For raspberry jam we took 12 kg raspberries and 8 kg Sahara. How much sugar will you need if you took it? 9 kg raspberries?
Solution.
We reason like this: let it be necessary x kg sugar for 9 kg raspberries The mass of raspberries and the mass of sugar are directly proportional quantities: how many times less raspberries are, the same number of times less sugar is needed. Therefore, the ratio of raspberries taken (by weight) ( 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 need to be taken 6 kg Sahara.
The solution of the problem It could be done like this:
Let on 9 kg raspberries need to be taken x kg Sahara.
(The arrows in the figure are directed in one direction, and up or down does not matter. Meaning: how many times the number 12 more number 9 , the same number of times 8 more number X, i.e. there is a direct relationship here).
Answer: on 9 kg I need to take some raspberries 6 kg Sahara.
Task 2. Car for 3 hours traveled the distance 264 km. How long will it take him to travel? 440 km, if he drives at the same speed?
Solution.
Let for x hours the car will cover the distance 440 km.
Answer: the car will pass 440 km in 5 hours.
Task 3. Water flows from the pipe into the pool. Behind 2 hours she fills 1/5 swimming pool What part of the pool is filled with water in 5 o'clock?
Solution.
We answer the question of the task: for 5 o'clock will be filled 1/x part of the pool. (The entire pool is taken as one whole).