Basic goals:
- introduce the concept of direct and inverse proportional dependence of quantities;
- teach how to solve problems using these dependencies;
- promote the development of problem solving skills;
- consolidate the skill of solving equations using proportions;
- repeat the steps with ordinary and decimal fractions;
- develop logical thinking students.
DURING THE CLASSES
I. Self-determination for activity(Organizing time)
- Guys! Today in the lesson we will get acquainted with problems solved using proportions.
II. Updating knowledge and recording difficulties in activities
2.1. Oral work (3 min)
– Find the meaning of the expressions and find out the word encrypted in the answers.
14 – s; 0.1 – and; 7 – l; 0.2 – a; 17 – in; 25 – to
– The resulting word is strength. Well done!
– The motto of our lesson today: Power is in knowledge! I'm searching - that means I'm learning!
– Make up a proportion from the resulting numbers. (14:7 = 0.2:0.1 etc.)
2.2. Let's consider the relationship between the quantities we know (7 min)
– the distance covered by the car at a constant speed, and the time of its movement: S = v t ( with increasing speed (time), the distance increases);
– vehicle speed and time spent on the journey: v=S:t(as the time to travel the path increases, the speed decreases);
–
the cost of goods purchased at one price and its quantity:
C = a · n (with an increase (decrease) in price, the purchase cost increases (decreases));
– price of the product and its quantity: a = C: n (with an increase in quantity, the price decreases)
– area of the rectangle and its length (width): S = a · b (with increasing length (width), the area increases;
– rectangle length and width: a = S: b (as the length increases, 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 have obtained dependences in which, with an increase in one quantity several times, another immediately increases by the same amount (examples are shown with arrows) and dependences 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 proportionality.
Directly proportional dependence– a relationship in which as one value increases (decreases) several times, the second value increases (decreases) by the same amount.
Inversely proportional relationship– a relationship in which as one value increases (decreases) several times, the second value decreases (increases) by the same amount.
III. Setting a learning task
– What problem is facing us? (Learn to distinguish between direct and inverse dependencies)
- This - target our lesson. Now formulate topic lesson. (Direct and inverse proportional relationship).
- Well done! Write down the topic of the lesson in your notebooks. (The teacher writes the topic on the board.)
IV. "Discovery" of new knowledge(10 min)
Let's look at problem No. 199.
1. The printer prints 27 pages in 4.5 minutes. How long will it take it to print 300 pages?
27 pages – 4.5 min.
300 pages - x?
2. The box contains 48 packs of tea, 250 g each. How many 150g packs of this tea will you get?
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 clutch gears has 32 teeth, and the other has 40. How many revolutions will the second gear make while the first one makes 215 revolutions?
32 teeth – 315 rev.
40 teeth – x?
To compile a proportion, one direction of the arrows is necessary; for this, in inverse proportionality, one ratio is replaced by the inverse.
At the board, students find the meaning of quantities; on the spot, 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 consolidation in external speech(10 min)
Worksheet assignments:
- From 21 kg of cottonseed, 5.1 kg of oil was obtained. How much oil will be obtained from 7 kg of cottonseed?
- To build the stadium, 5 bulldozers cleared the site in 210 minutes. How long would it take 7 bulldozers to clear this site?
VI. Independent work with self-test against standard(5 minutes)
Two students complete tasks No. 225 independently on hidden boards, and the rest are in notebooks. They then check the algorithm's work and compare it with the solution on the board. Errors are corrected and their causes are determined. If the task is completed correctly, then the students put a “+” sign next to them.
Students who make mistakes in independent work can use consultants.
VII. Inclusion in the knowledge system and repetition№ 271, № 270.
Six people work at the board. After 3-4 minutes, students working at the board present their solutions, and the rest check the assignments and participate in their discussion.
VIII. Reflection on activity (lesson summary)
– What new did you learn in the lesson?
-What did they repeat?
– What is the algorithm for solving proportion problems?
– Have we achieved our goal?
– How do you evaluate your work?
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.
- Newton's second law
- Coulomb barrier
See what “Direct proportionality” is in other dictionaries:
direct proportionality- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Energy topics in general EN direct ratio ... Technical Translator's Guide
direct proportionality- tiesioginis proporcingumas statusas T sritis fizika atitikmenys: engl. direct proportionality vok. direkte Proportionalität, f rus. direct proportionality, f pranc. proportionnalité directe, f … Fizikos terminų žodynas
PROPORTIONALITY- (from Latin proportionalis proportionate, proportional). Proportionality. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. PROPORTIONALITY lat. proportionalis, proportional. Proportionality. Explanation 25000... ... Dictionary of foreign words of the Russian language
PROPORTIONALITY- PROPORTIONALITY, proportionality, plural. no, female (book). 1. abstract noun to proportional. Proportionality of parts. Body proportionality. 2. Such a relationship between 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, female. 1. see proportional. 2. In mathematics: such a relationship between quantities in which an increase in one of them entails a change in the other by the same amount. Straight line (with a cut with an increase in one value... ... Ozhegov's Explanatory Dictionary
proportionality- And; and. 1. to Proportional (1 value); proportionality. P. parts. P. physique. P. representation in parliament. 2. Math. Dependence between proportionally changing quantities. Proportionality factor. Direct line (in which with... ... encyclopedic Dictionary
g) the age of the person and the size of his shoes;
h) the volume of the cube and the length of its edge;
i) the perimeter of the square and the length of its side;
j) a fraction and its denominator, if the numerator does not change;
k) a fraction and its numerator if the denominator does not change.
Solve problems 767-778 by composing.
767. A steel ball with a volume of 6 cm 3 has a mass of 46.8 g. What is the mass of a ball made of the same steel if its volume is 2.5 cm 3?
768. From 21 kg of cotton seed, 5.1 kg of oil was obtained. How much oil will be obtained from 7 kg of cottonseed?
769. 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?
770. To transport the cargo, 24 vehicles with a lifting capacity of 7.5 tons were required. How many vehicles with a lifting capacity of 4.5 tons are needed to transport the same cargo?
771. To determine the germination of seeds, peas were sown. Of the 200 peas sown, 170 sprouted. What percentage of the peas sprouted (germination percentage)?
772. During the city greening Sunday, linden trees were planted on the street. 95% of all planted linden trees were accepted. How many linden trees were planted if 57 linden trees were planted?
773. There are 80 students in the ski section. Among them are 32 girls. Which section members are girls and which are boys?
774. According to the plan, the collective farm should sow 980 hectares with corn. But the plan was fulfilled by 115%. How many hectares of corn did the collective farm sow?
775. In 8 months, the worker completed 96% of the annual plan. What percentage of the annual plan will the worker complete in 12 months if he works with the same productivity?
776. In three days, 16.5% of all beets were harvested. How many days will it take to harvest 60.5% of all beets at the same productivity?
777.V iron ore For 7 parts of iron there are 3 parts of impurities. How many tons of impurities are in the ore that contains 73.5 tons of iron?
778. To prepare borscht, for every 100 g of meat you need to take 60 g of beets. How many beets should you take for 650 g of meat?
P 779. Calculate orally:
780. Present each of the following fractions as the sum of two fractions with numerator 1: 
.
781. From the numbers 3, 7, 9 and 21, form two correct proportions.
782. The middle terms of the proportion are 6 and 10. What can the extreme terms be? Give examples.
783. At what value of x is the proportion correct:
784. Find the relation:
a) 2 min to 10 s; c) 0.1 kg to 0.1 g; e) 3 dm 3 to 0.6 m 3.
b) 0.3 m 2 to 0.1 dm 2; d) 4 hours to 1 day;
1) 6,0008:2,6 + 4,23 0,4;
2) 2,91 1,2 + 12,6288:3,6.
D 795. 20 kg of apples yield 16 kg of applesauce. ^^ How much applesauce will you get from 45 kg of apples?
796. Three painters can finish the job in 5 days. To speed up the work, two more painters were added. How long will it take them to finish the job, assuming that all painters will work with the same productivity?
797. For 2.5 kg of lamb they paid 4.75 rubles. How much lamb can you buy at the same price for 6.65 rubles?
798. Sugar beets contain 18.5% sugar. How much sugar is contained in 38.5 tons of sugar beets? Round your answer to tenths of a ton.
799. The new variety of sunflower seeds contain 49.5% oil. How many kilograms of these seeds must be taken so that they contain 29.7 kg of oil?
800. 80 kg of potatoes contain 14 kg of starch. Find the percentage of starch in such potatoes.
801. Flax seeds contain 47% oil. How much oil is contained in 80 kg of flax seeds?
802. Rice contains 75% starch, and barley 60%. How much barley do you need to take so that it contains the same amount of starch as is contained in 5 kg of rice?
803. Find the meaning of the expression:
a) 203.81:(141 -136.42) + 38.4:0.7 5;
b) 96:7.5 + 288.51:(80 - 76.74).
N.Ya.Vilenkin, A.S. Chesnokov, S.I. Shvartsburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school
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 a numerical expression for everything in the world. They discovered; What mathematical proportions lie at the basis of 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, they 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 of life, to 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 terms of proportion, 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 middle term of a proportion, 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 you can see, problems involving 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
IN school library 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 fills the 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 oldest sciences; it arose on the basis of the needs and wants 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 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 (at 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.