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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.
Solution.
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
Solution.
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.
Today we will look at what quantities are called inversely proportional, what the inverse proportionality graph looks like, and how all this can be useful to you not only in mathematics lessons, but also outside the school walls.
Such different proportions
Proportionality name two quantities that are mutually dependent on each other.
Dependence can be direct and reverse. Therefore, the relationship between quantities describe direct and inverse proportionality.
Direct proportionality- this is such a relationship between two quantities, in which an increase or decrease in one of them leads to an increase or decrease in the other. Those. their attitude does not change.
For example, the more effort you put into preparing for exams, the higher your grades will be. Or the more things you take with you on a hike, the harder it is to carry your backpack. Those. the amount of effort spent on preparing for exams is directly proportional to the grades received. And the number of things packed in a backpack is directly proportional to its weight.
Inverse proportionality- this is a functional dependence in which a decrease or increase by several times independent value(it is called an argument) causes a proportional (i.e., the same number of times) increase or decrease in the dependent value (it is called a function).
Illustrate simple example. You want to buy apples in the market. The apples on the counter and the amount of money in your wallet are inversely related. Those. the more apples you buy, the less money you will have left.
Function and its graph
The inverse proportionality function can be described as y = k/x. Wherein x≠ 0 and k≠ 0.
This function has the following properties:
- Its domain of definition is the set of all real numbers except x = 0. D(y): (-∞; 0) U (0; +∞).
- The range is all real numbers except y= 0. E(y): (-∞; 0) U (0; +∞) .
- It has no maximum or minimum values.
- Is odd and its graph is symmetrical about the origin.
- Non-periodic.
- Its graph does not cross the coordinate axes.
- Has no zeros.
- If a k> 0 (that is, the argument increases), the function decreases proportionally on each of its intervals. If a k< 0 (т.е. аргумент убывает), функция пропорционально возрастает на каждом из своих промежутков.
- As the argument increases ( k> 0) the negative values of the function are in the interval (-∞; 0), and the positive values are in the interval (0; +∞). When the argument is decreasing ( k< 0) отрицательные значения расположены на промежутке (0; +∞), положительные – (-∞; 0).
The graph of the inverse proportionality function is called a hyperbola. Depicted as follows:
Inverse Proportional Problems
To make it clearer, let's look at a few tasks. They are not too complicated, and their solution will help you visualize what inverse proportion is and how this knowledge can be useful in your everyday life.
Task number 1. The car is moving at a speed of 60 km/h. It took him 6 hours to reach his destination. How long will it take him to cover the same distance if he moves at twice the speed?
We can start by writing down a formula that describes the relationship of time, distance and speed: t = S/V. Agree, it very much reminds us of the inverse proportionality function. And it indicates that the time that the car spends on the road, and the speed with which it moves, are inversely proportional.
To verify this, let's find V 2, which, by condition, is 2 times higher: V 2 \u003d 60 * 2 \u003d 120 km / h. Then we calculate the distance using the formula S = V * t = 60 * 6 = 360 km. Now it is not difficult to find out the time t 2 that is required from us according to the condition of the problem: t 2 = 360/120 = 3 hours.
As you can see, travel time and speed are indeed inversely proportional: with a speed 2 times higher than the original one, the car will spend 2 times less time on the road.
The solution to this problem can also be written as a proportion. Why do we create a diagram like this:
↓ 60 km/h – 6 h
↓120 km/h – x h
Arrows indicate an inverse relationship. And they also suggest that when drawing up the proportion, the right side of the record must be turned over: 60/120 \u003d x / 6. Where do we get x \u003d 60 * 6/120 \u003d 3 hours.
Task number 2. The workshop employs 6 workers who cope with a given amount of work in 4 hours. If the number of workers is halved, how long will it take for the remaining workers to complete the same amount of work?
We write the conditions of the problem in the form of a visual diagram:
↓ 6 workers - 4 hours
↓ 3 workers - x h
Let's write this as a proportion: 6/3 = x/4. And we get x \u003d 6 * 4/3 \u003d 8 hours. If there are 2 times fewer workers, the rest will spend 2 times more time to complete all the work.
Task number 3. Two pipes lead to the pool. Through one pipe, water enters at a rate of 2 l / s and fills the pool in 45 minutes. Through another pipe, the pool will be filled in 75 minutes. How fast does water enter the pool through this pipe?
To begin with, we will bring all the quantities given to us according to the condition of the problem to the same units of measurement. To do this, we express the filling rate of the pool in liters per minute: 2 l / s \u003d 2 * 60 \u003d 120 l / min.
Since it follows from the condition that the pool is filled more slowly through the second pipe, it means that the rate of water inflow is lower. On the face of inverse proportion. Let us express the speed unknown to us in terms of x and draw up the following scheme:
↓ 120 l/min - 45 min
↓ x l/min – 75 min
And then we will make a proportion: 120 / x \u003d 75/45, from where x \u003d 120 * 45/75 \u003d 72 l / min.
In the problem, the filling rate of the pool is expressed in liters per second, let's bring our answer to the same form: 72/60 = 1.2 l/s.
Task number 4. Business cards are printed in a small private printing house. An employee of the printing house works at a speed of 42 business cards per hour and works full time - 8 hours. If he worked faster and printed 48 business cards per hour, how much sooner could he go home?
We go in a proven way and draw up a scheme according to the condition of the problem, denoting the desired value as x:
↓ 42 business cards/h – 8 h
↓ 48 business cards/h – xh
Before us back proportional dependence: how many times more business cards an employee of a printing house prints per hour, the same amount of time it will take him to complete the same job. Knowing this, we can set up the proportion:
42/48 \u003d x / 8, x \u003d 42 * 8/48 \u003d 7 hours.
Thus, having completed the work in 7 hours, the printing house employee could go home an hour earlier.
Conclusion
It seems to us that these inverse proportionality problems are really simple. We hope that now you also consider them so. And most importantly, knowledge of the inversely proportional dependence of quantities can really be useful to you more than once.
Not only in math classes and exams. But even then, when you are going to go on a trip, go shopping, decide to earn some money during the holidays, etc.
Tell us in the comments what examples of inverse and direct proportionality you notice around you. Let this be a game. You'll see how exciting it is. Don't forget to share this article in social networks so that your friends and classmates can also play.
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The concept of direct proportionality
Imagine that you are thinking of buying your favorite candy (or whatever you really like). The sweets in the store have their own price. Suppose 300 rubles per kilogram. The more candies you buy, the more money pay. That is, if you want 2 kilograms - pay 600 rubles, and if you want 3 kilos - give 900 rubles. Everything seems to be clear with this, right?
If yes, then it is now clear to you what direct proportionality is - this is a concept that describes the ratio of two quantities that depend on each other. And the ratio of these quantities remains unchanged and constant: by how many parts one of them increases or decreases, by the same number of parts the second increases or decreases proportionally.
Direct proportionality can be described by the following formula: f(x) = a*x, and a in this formula is a constant value (a = const). In our candy example, the price is a constant, a constant. It does not increase or decrease, no matter how many sweets you decide to buy. The independent variable (argument) x is how many kilograms of sweets you are going to buy. And the dependent variable f(x) (function) is how much money you end up paying for your purchase. So we can substitute the numbers in the formula and get: 600 r. = 300 r. * 2 kg.
The intermediate conclusion is this: if the argument increases, the function also increases, if the argument decreases, the function also decreases
Function and its properties
Direct proportional function is a special case of a linear function. If the linear function is y = k*x + b, then for direct proportionality it looks like this: y = k*x, where k is called the proportionality factor, and this is always a non-zero number. Calculating k is easy - it is found as a quotient of a function and an argument: k = y/x.
To make it clearer, let's take another example. Imagine that a car is moving from point A to point B. Its speed is 60 km/h. If we assume that the speed of movement remains constant, then it can be taken as a constant. And then we write the conditions in the form: S \u003d 60 * t, and this formula is similar to the direct proportionality function y \u003d k * x. Let's draw a parallel further: if k \u003d y / x, then the speed of the car can be calculated, knowing the distance between A and B and the time spent on the road: V \u003d S / t.
And now, from the applied application of knowledge about direct proportionality, let's return back to its function. The properties of which include:
its domain of definition is the set of all real numbers (as well as its subset);
the function is odd;
the change in variables is directly proportional to the entire length of the number line.
Direct proportionality and its graph
A graph of a direct proportional function is a straight line that intersects the origin point. To build it, it is enough to mark only one more point. And connect it and the origin of the line.
In the case of a graph, k is the slope. If the slope is less than zero (k< 0), то угол между графиком функции прямой пропорциональности и осью абсцисс тупой, а функция убывающая. Если угловой коэффициент больше нуля (k >0), the graph and the x-axis form an acute angle, and the function is increasing.
And one more property of the graph of the direct proportionality function is directly related to the slope k. Suppose we have two non-identical functions and, accordingly, two graphs. So, if the coefficients k of these functions are equal, their graphs are parallel on the coordinate axis. And if the coefficients k are not equal to each other, the graphs intersect.
Task examples
Let's decide a couple direct proportionality problems
Let's start simple.
Task 1: Imagine that 5 hens laid 5 eggs in 5 days. And if there are 20 hens, how many eggs will they lay in 20 days?
Solution: Denote the unknown as x. And we will argue as follows: how many times have there been more chickens? Divide 20 by 5 and find out that 4 times. And how many times more eggs will 20 hens lay in the same 5 days? Also 4 times more. So, we find ours like this: 5 * 4 * 4 \u003d 80 eggs will be laid by 20 hens in 20 days.
Now the example is a little more complicated, let's rephrase the problem from Newton's "General Arithmetic". Task 2: A writer can write 14 pages of a new book in 8 days. If he had assistants, how many people would it take to write 420 pages in 12 days?
Solution: We reason that the number of people (writer + assistants) increases with the increase in the amount of work if it had to be done in the same amount of time. But how many times? Dividing 420 by 14, we find out that it increases by 30 times. But since, according to the condition of the task, more time is given for work, the number of assistants does not increase by 30 times, but in this way: x \u003d 1 (writer) * 30 (times): 12/8 (days). Let's transform and find out that x = 20 people will write 420 pages in 12 days.
Let's solve another problem similar to those that we had in the examples.
Task 3: Two cars set off on the same journey. One was moving at a speed of 70 km/h and covered the same distance in 2 hours as the other in 7 hours. Find the speed of the second car.
Solution: As you remember, the path is determined through speed and time - S = V *t. Since both cars traveled the same way, we can equate the two expressions: 70*2 = V*7. Where do we find that the speed of the second car is V = 70*2/7 = 20 km/h.
And a couple more examples of tasks with direct proportionality functions. Sometimes in problems it is required to find the coefficient k.
Task 4: Given the functions y \u003d - x / 16 and y \u003d 5x / 2, determine their proportionality coefficients.
Solution: As you remember, k = y/x. Hence, for the first function, the coefficient is -1/16, and for the second, k = 5/2.
And you may also come across a task like Task 5: Write down the direct proportionality formula. Its graph and the graph of the function y \u003d -5x + 3 are located in parallel.
Solution: The function that is given to us in the condition is linear. We know that direct proportionality is a special case of a linear function. And we also know that if the coefficients of k functions are equal, their graphs are parallel. This means that all that is required is to calculate the coefficient of a known function and set direct proportionality using the familiar formula: y \u003d k * x. Coefficient k \u003d -5, direct proportionality: y \u003d -5 * x.
Conclusion
Now you have learned (or remembered, if you have already covered this topic before), what is called direct proportionality, and considered it examples. We also talked about the direct proportionality function and its graph, solved a few problems for example.
If this article was useful and helped to understand the topic, tell us about it in the comments. So that we know if we could benefit you.
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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 students' logical thinking.
DURING THE CLASSES
I. Self-determination to activity(Organizing time)
- Guys! Today in the lesson we will get acquainted with the problems solved using proportions.
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 - s; 0.1 - and; 7 - l; 0.2 - a; 17 - in; 25 - to
- The word came out - strength. Well done!
- The motto of our lesson today: Power is in knowledge! I'm looking - so I'm learning!
- Make a proportion of the resulting numbers. (14:7=0.2:0.1 etc.)
2.2. Consider the relationship between known quantities (7 min)
- 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 road: v=S:t(with an increase in the time to travel the path, the speed decreases);
–
the cost of goods purchased at one price and its quantity:
C \u003d a n (with an increase (decrease) in price, the cost of purchase increases (decreases);
- the price of the product and its quantity: a \u003d C: n (with an increase in quantity, the price decreases)
- the area of the rectangle and its length (width): S = a b (with an increase in the length (width), the area increases;
- the length of the rectangle and the width: a = S: b (with an increase in the 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 \u003d A: n (with an increase in the number of workers, the time spent on doing work decreases), etc.
We have obtained dependencies in which, with an increase in one value several times, the other immediately increases by the same amount (shown with arrows for examples) and dependencies in which, with an increase in one value several times, the second value decreases by the same number of times.
Such relationships are called direct and inverse proportions.
Directly proportional dependence- a dependence in which with an increase (decrease) in one value several times, the second value increases (decreases) by the same amount.
Inverse proportional relationship- a dependence in which with an increase (decrease) in one value several times, the second value decreases (increases) by the same amount.
III. Statement of the learning task
What is the problem we are facing? (Learn to distinguish between direct and inverse relationships)
- It - goal our lesson. Now formulate topic lesson. (Direct and inverse proportionality).
- Well done! Write the topic of the lesson in your notebooks. (The teacher writes the topic on the blackboard.)
IV. "Discovery" of new knowledge(10 min)
Let's analyze problems number 199.
1. The printer prints 27 pages in 4.5 minutes. How long will it take to print 300 pages?
27 pages - 4.5 min.
300 pp. - 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, having spent 25 liters of gasoline. How far can a car travel on a full tank of 40 liters?
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 will make 215 revolutions?
32 teeth - 315 rpm
40 teeth - x?
To draw up a proportion, one direction of the arrows is necessary, for this, in inverse proportion, one ratio is replaced by the inverse.
At the blackboard, students find the value of the quantities, in the field, students solve one problem of their choice.
– Formulate a rule for solving problems with direct and inverse proportionality.
A table appears on the board:
V. Primary consolidation in external speech(10 min)
Tasks on the sheets:
- From 21 kg of cottonseed, 5.1 kg of oil was obtained. How much oil will be obtained 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. Independent work with self-test according to the standard(5 minutes)
Two students complete assignments No. 225 on their own on hidden boards, and the rest in notebooks. Then they check the work according to the algorithm and compare it with the solution on the board. Errors are corrected, their causes are clarified. If the task is completed, right, then next to the students put a “+” sign for themselves.
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 blackboard. After 3–4 minutes, the students who worked at the blackboard present their solutions, and the rest check the tasks and participate in their discussion.
VIII. Reflection of activity (the result of the lesson)
- What new did you learn at the lesson?
- What did you repeat?
What is the algorithm for solving proportion problems?
Have we reached our goal?
- How do you rate your work?
I. Directly proportional quantities.
Let the value y depends on the size X. If with an increase X several times the size at increases by the same factor, then such values X and at are called directly proportional.
Examples.
1 . The quantity of the purchased goods and the cost of the purchase (at a fixed price of one unit of goods - 1 piece or 1 kg, etc.) How many times more goods were bought, so many times more and paid.
2 . The distance traveled and the time spent on it (at constant speed). How many times longer the path, how many times more time we will spend on 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 quantities.
If two quantities are directly proportional, then the ratio of two arbitrary values of the first quantity is equal to the ratio of the two corresponding values of the second quantity.
Task 1. For raspberry jam 12 kg raspberries and 8 kg Sahara. How much sugar will be required if taken 9 kg raspberries?
Solution.
We argue like this: let it be necessary x kg sugar on 9 kg raspberries. The mass of raspberries and the mass of sugar are directly proportional: how many times less raspberries, the same amount of sugar is needed. Therefore, the ratio of taken (by weight) raspberries ( 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 the 9 kg raspberries to take 6 kg Sahara.
The solution of the problem could have been done like this:
Let on 9 kg raspberries to take x kg Sahara.
(The arrows in the figure are directed in one direction, and it does not matter up or down. Meaning: how many times the number 12 more number 9 , the same number 8 more number X, i.e., there is a direct dependence here).
Answer: on the 9 kg raspberries to take 6 kg Sahara.
Task 2. car for 3 hours traveled distance 264 km. How long will it take him 440 km if it travels 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.
