I. Directly proportional values.
Let the value y depends on the value X... If when increasing X several times the magnitude at increases by the same factor, then such values X and at are called directly proportional.
Examples.
1 ... The quantity of purchased goods and the cost of purchase (at a fixed price of one unit of goods - 1 piece or 1 kg, etc.) How many times more goods were bought, how many times more they paid.
2 ... Distance traveled and time spent on it (for constant speed).How many times the path is longer, so many times more time will be spent to walk 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 values.
If two quantities are directly proportional, then the ratio of two arbitrary values of the first quantity is equal to the ratio of two corresponding values of the second quantity.
Objective 1. For raspberry jam have taken 12 Kg raspberries and 8 Kg Sahara. How much sugar is required if taken 9 Kg raspberries?
Solution.
We reason like this: let it be required x kg sugar on 9 Kg raspberries. The mass of raspberries and the mass of sugar are directly proportional values: how many times less than raspberries, the same times less sugar is needed. Therefore, the ratio of taken (by weight) raspberries ( 12:9 ) will be equal to the ratio of the taken sugar ( 8: x). We get the proportion:
12: 9=8: X;
x = 9 · 8: 12;
x = 6. Answer: on the 9 Kg raspberries need to be taken 6 Kg Sahara.
The solution of the problem could have been arranged like this:
Let on 9 Kg raspberries need to be taken x kg Sahara.
(The arrows in the figure are directed in one direction, but up or down does not matter. Meaning: how many times the number 12 more numbers 9 , the same number of times 8 more numbers X, i.e., there is a direct relationship).
Answer: on the 9 Kg raspberries need to be taken 6 Kg Sahara.
Objective 2. Car for 3 hours drove the distance 264 km... How long will it take 440 km if it 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.
The two quantities are called directly proportional, if when one of them is increased by several times, the other is increased by the same amount. Accordingly, when one of them decreases several times, the other decreases by the same amount.
The relationship between such values is a direct proportional relationship. Examples of direct proportional relationship:
1) at a constant speed, the distance traveled is directly proportional to time;
2) the perimeter of the square and its side are directly proportional values;
3) the cost of a product purchased at one price is directly proportional to its quantity.
To distinguish direct proportional dependence from inverse, you can use the proverb: "The further into the forest, the more firewood."
It is convenient to solve problems with directly proportional quantities using proportion.
1) To make 10 parts, you need 3.5 kg of metal. How much metal will be used to make 12 of these parts?
(We reason like this:
1. In the filled column, put the arrow in the direction from more to less.
2. The more parts, the more metal is 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 the proportion (in the direction from the beginning of the arrow to its end):
12: 10 = x: 3.5
To find, it is necessary 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) 1,680 rubles were paid for 15 meters of fabric. How much does 12 meters of such fabric cost?
(1. In the filled column, put the arrow in the direction from the largest number to the smallest.
2. The less fabrics are bought, the less you have to pay for them. This means that this is a directly proportional relationship.
3. Therefore, the second arrow is in the same direction with the first).
Let x rubles cost 12 meters of fabric. We make the proportion (from the beginning of the arrow to its end):
15: 12 = 1680: x
To find the unknown extreme term of the proportion, we divide the product of the middle terms by the known extreme term of the proportion:
This means that 12 meters cost 1,344 rubles.
Answer: 1344 rubles.
Completed by: Chepkasov Rodion
student of 6 "B" grade
MBOU "Secondary School No. 53"
Barnaul
Head: Bulykina O.G.
mathematic teacher
MBOU "Secondary School No. 53"
Barnaul
Introduction. one
Relationships and proportions. 3
Direct and inverse proportional relationships. 4
Application of direct and inverse proportional 6
dependencies in solving various problems.
Conclusion. eleven
Literature. 12
Introduction.
The word proportion comes from the Latin word proportion, meaning in general 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 deep antiquity, man discovered that all phenomena in nature are connected with each other, that everything is in constant motion, change, and, being expressed by a number, reveals amazing regularities.
The Pythagoreans and their followers were looking for everything in the world numeric expression... It was discovered by them; that mathematical proportions are at the heart of music (the ratio of string length to 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, argued that symmetrical geometric shapes... The Pythagoreans were looking for 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, proportionality, however, first of all exists in numbers. It is necessary that everything be numbered." 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 laws hidden in nature that a scientist knows in the form of a numerical law."
Proportions were used in solving various problems both in antiquity and in the Middle Ages. Certain types of problems are now easily and quickly solved using proportions. Proportions and proportionality have been and are applied not only in mathematics, but also in architecture and art. Proportionality in architecture and art means the observance of certain proportions between the dimensions of different parts of a building, figure, sculpture or other work of art. Proportionality in such cases is a condition for correct and beautiful construction and image.
In my work, I tried to consider the application of direct and inverse proportional dependencies in different areas surrounding life, trace the connection with academic subjects through tasks.
Relationships and proportions.
The quotient of two numbers is called attitude of these numbers.
Attitude shows, how many times the first number is greater than the second, or how much of the first number is from the second.
Task.
2.4 tons of pears and 3.6 tons of apples were brought to the store. What part of the imported fruits are pears?
Solution ... Let's find how many fruits were brought in: 2.4 + 3.6 = 6 (t). To find what part of the imported fruits are pears, let's compose the ratio 2.4: 6 =. The answer can also be written as a decimal fraction or as a percentage: = 0.4 = 40%.
Mutually inverse are called the numbers whose products are equal to 1. Therefore a relationship is called an inverse 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 ratios: a: b = c: d or = 
, where a and d are extreme terms of proportion, c and b - middle members(all members of the proportion are nonzero).
The main property of proportion:
in the correct proportion, the product of the extreme terms is equal to the product of the middle terms.
Applying the displacement property of multiplication, we get that the extreme terms or middle terms can be interchanged in the correct proportion. The resulting proportions will also be correct.
Using the main property of proportion, you can find its unknown term if all other terms are known.
To find the unknown extreme term of the proportion, it is necessary to multiply the middle terms and divide by the known extreme term. x: b = c: d, x = 
To find the unknown average term of the proportion, it is necessary to multiply the extreme terms and divide by the known average 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 called proportional if, with increasing
(decrease) one of them several times, the other increases (decreases) by the same amount.
If two quantities are directly proportional, then the ratios of the corresponding values of these quantities are equal.
Example direct proportional relationship .
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 ratio of weight to volume 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 the quantities are inversely proportional, then the ratio of the values of one quantity is equal to the inverse ratio of the corresponding values of the other quantity.
P exampleinverse proportional relationship.
The 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. Let's find the width of the second rectangle.
Solution:
1 rectangle 3.6 m 2.4 m
2 rectangle 4.8 mx m
3.6 mx m
4.8 m 2.4 m
x = 3.6 * 2.4 = 1.8 m
Answer: 1.8 m.
As you can see, tasks for proportional values can be solved using proportions.
Not all two quantities are directly proportional or inversely proportional. For example, a child's height increases with increasing age, but these values are not proportional, since when the age is doubled, the child's height does not double.
Practical use direct and inverse proportional dependence.
Problem number 1
The school library has 210 mathematics textbooks, which is 15% of the total library fund. How many books are there in the library collection?
Solution:
Total textbooks -? - one hundred%
Mathematicians - 210 -15%
15% 210 account
X = 100 * 210 = 1400 textbooks
100% x account 15
Answer: 1400 textbooks.
Problem number 2
A cyclist travels 75 km in 3 hours. How long does it take for a cyclist to travel 125 km at the same speed?
Solution:
3 h - 75 km
H - 125 km
Time and distance are directly proportional, therefore
3: x = 75: 125,
x = 
,
x = 5.
Answer: in 5 hours.
Problem number 3
8 identical pipes fill the pool in 25 minutes. How many minutes will it take to fill a pool of 10 such pipes?
Solution:
8 pipes - 25 minutes
10 pipes -? minutes
The number of pipes is inversely proportional to the time, therefore
8: 10 = x: 25,
x = 
x = 20
Answer: in 20 minutes.
Problem number 4
A team of 8 workers completes the task in 15 days. How many workers will be able to complete the task in 10 days, 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, therefore
x: 8 = 15: 10,
x = 
,
x = 12.
Answer: 12 workers.
Problem number 5
From 5.6 kg of tomatoes, 2 liters of sauce are obtained. How many liters of sauce can you get 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 p.
Problem number 6
Coal was prepared for heating the school building for 180 days at a consumption rate
0.6 tons of coal per day. How many days will this stock last if you spend 0.5 tonnes 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 number 7
In iron ore, 7 parts of iron account for 3 parts of impurities. How many tons of impurities are in the ore, which 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 number 8
The car drove 500 km, using 35 liters of gasoline. How many liters of gasoline will it take to travel 420 km?
Solution:
Distance, km
Gasoline, l
The distance is directly proportional to the consumption of gasoline, therefore
500: 35 = 420: x,
x = 35 * 420: 500,
x = 29.4.
Answer: 29.4 L
Problem number 9
12 crucians were caught in 2 hours. How many crucians will be caught in 3 hours?
Solution:
The number of crucians does not depend on time. These quantities are neither directly proportional nor inversely proportional.
Answer: There is no answer.
Problem number 10
A mining company needs to purchase 5 new machines for a certain amount of money at a price of 12 thousand rubles per one. How many such cars can a company buy if the price for one car becomes 15 thousand rubles?
Solution:
Number of cars, pcs.
Price, thousand rubles
The number of cars is inversely proportional to the cost, therefore
5: x = 15: 12,
x = 5 * 12: 15,
x = 4.
Answer: 4 cars.
Problem number 11
In the town N, there is a store in square P, the owner of which is so strict that for being late he deducts 70 rubles from his salary for 1 delay per day. Two girls, Yulia and Natasha, work in one department. Their wage depends on the number of working days. Julia received 4,100 rubles in 20 days, and Natasha should have received more in 21 days, but she was late 3 days in a row. How many rubles will Natasha get?
Solution:
Working day
Salary, rub.
Julia
4100
Natasha
The salary is directly proportional to the number of working days, therefore
20: 21 = 4100: x,
x = 4305.
RUB 4305 Natasha should have received.
4305 - 3 * 70 = 4095 (rub.)
Answer: Natasha will receive 4095 rubles.
Problem number 12
The distance between two cities on the map is 6 cm. Find the distance between these cities on the terrain if the map scale is 1: 250000.
Solution:
Let's denote the distance between cities on the terrain through x (in centimeters) and find the ratio of the length of the segment on the map to the distance on the terrain, which will be equal to the scale of the map: 6: x = 1: 250000,
x = 6 * 250,000,
x = 1,500,000.
1500000 cm = 15 km
Answer: 15 km.
Problem number 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 number 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:
RUB 50,000
100%
x rub.
50,000: x = 100: 10,
x = 50,000 * 10: 100,
x = 5000.
RUB 5,000 is 10%.
50,000 + 5000 = 55,000 (rub.)
Answer: 55,000 rubles will be returned to the bank in a year.
Conclusion.
As you can see from the above examples, direct and inverse proportional relationships are applicable in various areas of life:
Economy,
Trade,
In production and industry,
Cooking,
Construction and architecture.
Sports,
Livestock,
Topography,
Physicists,
Chemistry, etc.
In Russian, there are also proverbs and sayings that establish direct and inverse dependencies:
As it comes around, it will respond.
The higher the stump, the higher the shadow.
The more people there are, the less oxygen.
And it's done, but stupidly.
Mathematics is one of the oldest sciences, it arose on the basis of the needs and requirements of mankind. 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 proportional dependence has been known since ancient times, since it was the laws of proportion that moved architects during any construction or creation of any sculpture.
The knowledge of proportions is widely used in all spheres of human life and activities - you cannot do without them when writing paintings (landscapes, still lifes, portraits, etc.), it is also widespread among architects and engineers - in general, it is hard to imagine the creation of anything - anything without using knowledge about proportions and their ratio.
Literature.
Mathematics-6, N. Ya. Vilenkin and others.
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, IV Baranova et al., M. "Enlightenment" 1988
Collection of problems and examples in mathematics, grades 5-6, N.A. Tereshin,
T.N. Tereshina, M. "Aquarium" 1997
Basic goals:
- introduce the concept of direct and inverse proportional dependence of quantities;
- teach to solve problems using these dependencies;
- promote the development of problem solving skills;
- consolidate the skill of solving equations using proportion;
- repeat actions with ordinary and decimal fractions;
- develop logical thinking students.
DURING THE CLASSES
I. Self-determination to activity(Organizing time)
- Guys! Today in the lesson we will get acquainted with the problems solved using proportion.
II. Updating knowledge and fixing difficulties in activities
2.1. Oral work (3 min)
- Find the meaning of expressions and find out the word encrypted in the answers.
14 - c; 0.1 - and; 7 - l; 0.2 - a; 17 - c; 25 - to
- The word turned out - power. Well done!
- The motto of our lesson today: Power is in knowledge! I am looking - then I am learning!
- Make a proportion of the resulting numbers. (14: 7 = 0.2: 0.1, etc.)
2.2. Consider the relationship between the quantities we know (7 minutes)
- the path traveled by the car at a constant speed, and the time of its movement: S = v t ( with an increase in speed (time), the path increases);
- the speed of the car and the time spent on the way: v = S: t(with an increase in the time to travel the path, the speed decreases);
–
the cost of the goods purchased at one price and its quantity:
С = а · n (with an increase (decrease) in the price, the purchase price increases (decreases));
- the prices of the goods and their quantity: a = C: n (with an increase in quantity, the price decreases)
- the area of the rectangle and its length (width): S = a b (with increasing length (width), the area increases;
- the length of the rectangle and the width: a = S: b (with increasing length, the width decreases;
- the number of workers performing some work with the same labor productivity, and the time it takes to complete this work: t = A: n (with an increase in the number of workers, the time spent on performing the work decreases), etc.
We obtained dependencies in which, with an increase in one quantity several times, the other immediately increases by the same amount (show examples with arrows) and dependencies in which, with an increase in one quantity several times, the second quantity decreases by the same number of times.
Such dependencies are called direct and inverse proportions.
Directly proportional relationship- a dependence in which, with an increase (decrease) in one quantity several times, the second quantity increases (decreases) by the same amount.
Inversely proportional relationship- a dependence in which, with an increase (decrease) in one value by several times, the second value decreases (increases) by the same amount.
III. Staging learning task
- What problem faced us? (Learn to distinguish between direct and inverse dependencies)
- This - purpose our lesson. Now formulate theme lesson. (Direct and inverse proportional relationship).
- Well done! Write the lesson topic in your notebooks. (The teacher writes the topic on the chalkboard.)
IV. "Discovery" of new knowledge(10 min)
Let's look at problems # 199.
1. The printer prints 27 pages in 4.5 minutes. How long does it take to print 300 pages?
27 pages - 4.5 minutes
300 pages - x?
2. There are 48 packs of tea in a box, 250 g each. How many packs of 150g will come out of this tea?
48 packs - 250 g.
X? - 150 g.
3. The car drove 310 km using 25 liters of gasoline. How far can a car travel on a full 40L tank?
310 km - 25 l
X? - 40 l
4. One of the engaging gears has 32 teeth and the other has 40. How many revolutions will the second gear make while the first will make 215 revolutions?
32 teeth - 315 vol.
40 teeth - x?
To draw up the proportion, one direction of the arrows is necessary, for this, in inverse proportionality, one ratio is replaced by the opposite.
At the blackboard, students find the value of the quantities, on the ground, students solve one problem of their choice.
- Formulate a rule for solving problems with direct and inverse proportional dependence.
A table appears on the board:
V. Primary reinforcement in external speech(10 min)
Tasks on sheets:
- From 21 kg of cottonseed, 5.1 kg of oil was obtained. How much oil will be made from 7 kg of cottonseed?
- For the construction of the stadium, 5 bulldozers cleared the site in 210 minutes. How long would it take 7 bulldozers to clear this area?
Vi. Self-study with self-test by reference(5 minutes)
Two students complete assignments number 225 on their own on hidden boards, and the rest - in notebooks. Then they check the algorithm work and compare it with the solution on the board. Errors are corrected, their reasons are found out. If the task is completed correctly, then next to the students put themselves a "+" sign.
Students who make mistakes in independent work can use counselors.
Vii. Knowledge inclusion and repetition№ 271, № 270.
Six people work at the blackboard. After 3-4 minutes, the students who worked at the blackboard present their solutions, and the rest check the assignments and participate in their discussion.
VIII. Reflection of activity (lesson summary)
- What new have you learned in the lesson?
- What did you repeat?
- What is the algorithm for solving proportional problems?
- Have we reached our goal?
- How do you rate your work?
Example
1.6 / 2 = 0.8; 4/5 = 0.8; 5.6 / 7 = 0.8, etc.Aspect ratio
The constant ratio of proportional quantities is called aspect ratio... The proportionality coefficient shows how many units of one quantity fall on the 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 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 proportion
Inverse proportionality is a functional dependence in which an increase in the independent quantity (argument) causes a proportional decrease in the dependent quantity (function).
Mathematically, inverse proportionality is written as a formula:
Function properties:
Sources of
Wikimedia Foundation. 2010.
- Newton's second law
- Coulomb barrier
See what "Direct proportionality" is in other dictionaries:
direct proportion- - [A.S. Goldberg. The English Russian Energy Dictionary. 2006] Topics energy in general EN direct ratio ... Technical translator's guide
direct proportion- tiesioginis proporcingumas statusas T sritis fizika atitikmenys: angl. direct proportionality vok. direkte Proportionalität, f rus. direct proportionality, f pranc. proportionnalité directe, f… Fizikos terminų žodynas
PROPORTIONALITY- (from Lat. proportionalis proportional, proportional). Proportionality. Dictionary foreign words included in the Russian language. Chudinov AN, 1910. PROPORTIONALITY otlat. proportionalis, proportional. Proportionality. Explanation 25000 ... ... Dictionary of foreign words of the Russian language
PROPORTIONALITY- PROPORTIONALITY, proportionality, pl. no, wives. (book). 1.Distract. noun to proportional. Proportionality of parts. The proportionality of the physique. 2. Such a relationship between the quantities, when they are proportional (see proportional ... Dictionary Ushakova
Proportionality- Two mutually dependent quantities are called proportional if the ratio of their values remains unchanged .. Contents 1 Example 2 Proportionality coefficient ... Wikipedia
PROPORTIONALITY- PROPORTIONALITY, and, wives. 1. see proportional. 2. In mathematics: such a relationship between quantities, when a swarm of one of them increases, the other changes by the same amount. Straight p. (With a swarm with an increase in one value ... ... Ozhegov's Explanatory Dictionary
proportionality- and; f. 1. to Proportional (1 digit); proportionality. P. parts. P. physique. P. representation in parliament. 2. Mat. Relationship between proportionally varying quantities. Aspect ratio. Straight p. (In which with ... ... encyclopedic Dictionary