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 way, we will spend so many times more time to go through 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 have taken 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.
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.
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 decimals;
- 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 – c; 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 according to the standard(5 minutes)
Two students complete task No. 225 independently on hidden boards, and the rest - 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?
The two quantities are called directly proportional, if when one of them increases several times, the other increases by the same amount. Accordingly, when one of them decreases several times, the other decreases by the same amount.
The relationship between such quantities is a direct proportional relationship. Examples of direct proportional dependence:
1) at a constant speed, the distance traveled is directly proportional to time;
2) the perimeter of a square and its side are directly proportional quantities;
3) the cost of a product purchased at one price is directly proportional to its quantity.
To distinguish a direct proportional relationship from an inverse one, you can use the proverb: “The further into the forest, the more firewood.”
It is convenient to solve problems involving directly proportional quantities using proportions.
1) To make 10 parts you need 3.5 kg of metal. How much metal will go into making 12 of these parts?
(We reason like this:
1. In the filled column, place an arrow in the direction from more to less.
2. The more parts, the more metal 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 up the proportion (in the direction from the beginning of the arrow to its end):
12:10=x:3.5
To find , you need 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) For 15 meters of fabric they paid 1680 rubles. How much does 12 meters of such fabric cost?
(1. In the filled column, place an arrow in the direction from the largest number to the smallest.
2. The less fabric you buy, the less you have to pay for it. This means that this is a directly proportional relationship.
3. Therefore, the second arrow is in the same direction as the first).
Let x rubles cost 12 meters of fabric. We make a proportion (from the beginning of the arrow to its end):
15:12=1680:x
To find the unknown extreme term of the proportion, divide the product of the middle terms by the known extreme term of the proportion:
This means that 12 meters cost 1344 rubles.
Answer: 1344 rubles.