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 actions with ordinary and decimal fractions;
- 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 by several times, another 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)
- This - target 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?
In grades 7 and 8, a direct proportional graph is studied.
How to plot a direct proportional graph?
Consider an example of a direct proportionality graph.
Direct proportional graph formula
A direct proportional graph represents a function.
In general, direct proportionality has the formula
The slope of the direct proportionality graph with respect to the x-axis depends on the magnitude and sign of the direct proportionality coefficient.
The direct proportionality graph passes
The direct proportionality graph passes through the origin.
The direct proportionality graph is a straight line. The straight line is given by two points.
Thus, when constructing a graph of direct proportionality, it is enough to determine the position of two points.
But we always know one of them - this is the origin of coordinates.
It remains to find the second. Let's look at an example of constructing a graph of direct proportionality.
Plot the direct proportionality graph y = 2x
Task .
Plot the direct proportionality graph given by the formula
Solution .
All numbers are there.
We take any number from the area of \u200b\u200bdefinition of direct proportionality, let it be 1.
Find the value of the function when x is equal to 1
Y=2x=
2 * 1 = 2
that is, for x = 1 we get y = 2. The point with these coordinates belongs to the graph of the function y = 2x.
We know that a direct proportional graph is a straight line, and a straight line is given by two points.
The two quantities are called directly proportional, if when one of them is increased several times, the other is increased by the same amount. Accordingly, when one of them decreases by several times, the other decreases by the same amount.
The relationship between such quantities is a direct proportional relationship. Examples of a direct proportional relationship:
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;
3) the cost of a commodity 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 farther into the forest, the more firewood."
It is convenient to solve problems for directly proportional quantities using proportions.
1) For the manufacture of 10 parts, 3.5 kg of metal is needed. How much metal will be used to make 12 such parts?
(We argue like this:
1. In the completed column, put the arrow in the direction from the largest number to the smallest.
2. The more parts, the more metal is needed to make them. So it's 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 , we 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) 1680 rubles were paid for 15 meters of fabric. How much does 12 meters of such fabric cost?
(1. In the completed column, put the 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. So it's a directly proportional relationship.
3. Therefore, the second arrow is directed in the same direction as the first).
Let x rubles cost 12 meters of fabric. We make up the proportion (from the beginning of the arrow to its end):
15:12=1680:x
To find the unknown extreme member of the proportion, we divide the product of the middle terms by the known extreme member of the proportion:
So, 12 meters cost 1344 rubles.
Answer: 1344 rubles.
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 you 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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Example
1.6 / 2 = 0.8; 4 / 5 = 0.8; 5.6 / 7 = 0.8 etc.Proportionality factor
The constant ratio of proportional quantities is called coefficient of proportionality. The proportionality coefficient shows how many units of one quantity fall on a unit of another.
Direct proportionality
Direct proportionality- functional dependence, in which some 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 proportionality
Inverse proportion- 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
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