§ 129. Preliminary clarifications.

Man constantly deals with a wide variety of quantities. The employee and the worker try to get to the service, to work by a certain time, the pedestrian hurries to reach a certain place by the shortest route, the steam heating source worries that the temperature in the boiler is slowly rising, the business manager makes plans to reduce the cost of production, etc.

Any number of such examples could be cited. Time, distance, temperature, cost - all these are various quantities. In the first and second parts of this book, we got acquainted with some especially common quantities: area, volume, weight. We encounter many quantities in the study of physics and other sciences.

Imagine that you are on a train. From time to time you look at your watch and notice how long you have already been on the road. You say, for example, that 2, 3, 5, 10, 15 hours, etc. have elapsed since the departure of your train. These numbers indicate various periods of time; they are called values of this quantity (time). Or you look out the window and follow the road poles for the distance your train travels. The numbers 110, 111, 112, 113, 114 km flash before you. These numbers indicate the various distances that the train has traveled from the point of departure. They are also called values, this time with a different value (path or distance between two points). Thus, one value, for example, time, distance, temperature, can take on any different values.

Pay attention to the fact that a person almost never considers only one value, but always connects it with some other values. He has to deal simultaneously with two, three and more quantities. Imagine that you need to get to school by 9 o'clock. You look at your watch and see that you have 20 minutes. Then you quickly decide whether you should take the tram or you will have time to walk to the school. After thinking, you decide to walk. Note that at the time you were thinking, you were solving some problem. This task has become simple and familiar, as you solve such problems every day. In it, you quickly compared several values. It was you who looked at the clock, which means you took into account the time, then you mentally imagined the distance from your home to school; finally, you compared two quantities: the speed of your step and the speed of the tram, and concluded that in a given time (20 minutes) you will have time to walk. From this simple example, you can see that in our practice, some quantities are interconnected, that is, they depend on each other

In chapter twelve, it was told about the ratio of homogeneous quantities. For example, if one segment is 12 m and the other 4 m, then the ratio of these segments will be 12: 4.

We said that it is the ratio of two homogeneous quantities. In other words, it is the ratio of two numbers one name.

Now that we have become more familiar with quantities and have introduced the concept of the value of a quantity, we can state the definition of a relation in a new way. In fact, when we considered two segments of 12 m and 4 m, we were talking about one value - length, and 12 m and 4 m were only two different values of this value.

Therefore, in the future, when we start talking about a ratio, we will consider two values of one of some quantities, and the ratio of one value of a quantity to another value of the same quantity will be called the quotient of dividing the first value by the second.

§ 130. Quantities are directly proportional.

Consider a problem whose condition includes two quantities: distance and time.

Task 1. A body moving in a straight line and uniformly passes 12 cm in every second. Determine the path traveled by the body in 2, 3, 4, ..., 10 seconds.

Let's make a table by which it would be possible to monitor the change in time and distance.

The table gives us the opportunity to compare these two series of values. We see from it that when the values of the first quantity (time) gradually increase by 2, 3, ..., 10 times, then the values of the second quantity (distance) also increase by 2, 3, ..., 10 times. Thus, when the values of one quantity increase several times, the values of another quantity increase by the same amount, and when the values of one quantity decrease several times, the values of the other quantity decrease by the same amount.

Consider now a problem that includes two such quantities: the amount of matter and its cost.

Task 2. 15 m of fabric cost 120 rubles. Calculate the cost of this fabric for several other quantities of meters indicated in the table.

From this table, we can see how the value of a commodity gradually increases, depending on the increase in its quantity. Despite the fact that completely different quantities appear in this problem (in the first problem - time and distance, and here - the quantity of goods and its cost), nevertheless, a great similarity can be found in the behavior of these quantities.

Indeed, in the top line of the table are numbers indicating the number of meters of fabric, under each of them is written a number expressing the cost of the corresponding quantity of goods. Even a cursory glance at this table shows that the numbers in both the top and bottom rows are increasing; on closer examination of the table and when comparing individual columns, it turns out that in all cases the values of the second quantity increase by the same factor as the values of the first increase, i.e. if the value of the first quantity has increased, say, 10 times, then the value of the second value also increased 10 times.

If we look at the table from right to left, we will find that the indicated values \u200b\u200bof the quantities will decrease by the same number of times. In this sense, there is an unconditional similarity between the first task and the second.

The pairs of quantities that we met in the first and second problems are called directly proportional.

Thus, if two quantities are interconnected in such a way that with an increase (decrease) in the value of one of them several times, the value of the other increases (decreases) by the same amount, then such quantities are called directly proportional.

They also say about such quantities that they are interconnected by a directly proportional dependence.

In nature and in the life around us, there are many such quantities. Here are some examples:

1. Time work (a day, two days, three days, etc.) and earnings received during this time at day wages.

2. Volume any object made of a homogeneous material, and the weight this item.

§ 131. The property of directly proportional quantities.

Let's take a task that includes the following two quantities: working time and earnings. If the daily earnings are 20 rubles, then the earnings for 2 days will be 40 rubles, etc. It is most convenient to draw up a table in which a certain earnings will correspond to a certain number of days.

Looking at this table, we see that both quantities have taken 10 different values. Each value of the first value corresponds to a certain value of the second value, for example, 40 rubles correspond to 2 days; 5 days correspond to 100 rubles. In the table, these numbers are written one under the other.

We already know that if two quantities are directly proportional, then each of them, in the process of its change, increases by the same amount as the other increases. It immediately follows from this: if we take the ratio of any two values of the first quantity, then it will be equal to the ratio of the two corresponding values of the second quantity. Indeed:

Why is this happening? But because these values are directly proportional, that is, when one of them (time) increased by 3 times, then the other (earnings) increased by 3 times.

We have therefore come to the following conclusion: if we take any two values of the first magnitude and divide them one by the other, and then divide one by the other the corresponding values of the second magnitude, then in both cases one and the same number will be obtained, i.e., e. the same relation. This means that the two relations that we wrote above can be connected with an equal sign, i.e.

There is no doubt that if we took not these relations, but others, and not in that order, but in the opposite direction, we would also obtain equality of relations. Indeed, we will consider the values of our quantities from left to right and take the third and ninth values:

60:180 = 1 / 3 .

So we can write:

This implies the following conclusion: if two quantities are directly proportional, then the ratio of two arbitrarily taken values of the first quantity is equal to the ratio of the two corresponding values of the second quantity.

§ 132. Formula of direct proportionality.

Let's make a table of the cost of various quantities of sweets, if 1 kg of them costs 10.4 rubles.

Now let's do it this way. Let's take any number of the second row and divide it by the corresponding number of the first row. For example:

You see that in the quotient the same number is obtained all the time. Therefore, for a given pair of directly proportional quantities, the quotient of dividing any value of one quantity by the corresponding value of another quantity is a constant number (that is, not changing). In our example, this quotient is 10.4. This constant number is called the proportionality factor. In this case, it expresses the price of a unit of measurement, i.e., one kilogram of goods.

How to find or calculate the proportionality factor? To do this, you need to take any value of one quantity and divide it by the corresponding value of another.

Let us denote this arbitrary value of one quantity by the letter at , and the corresponding value of another quantity - the letter X , then the coefficient of proportionality (we denote it To) find by dividing:

In this equality at - divisible X - divider and To- quotient, and since, by the property of division, the dividend is equal to the divisor multiplied by the quotient, we can write:

y= K x

The resulting equality is called formula of direct proportionality. Using this formula, we can calculate any number of values of one of the directly proportional quantities, if we know the corresponding values \u200b\u200bof the other quantity and the coefficient of proportionality.

Example. From physics we know that the weight R of any body is equal to its specific gravity d multiplied by the volume of this body V, i.e. R = d V.

Take five iron ingots of various sizes; knowing the specific gravity of iron (7.8), we can calculate the weights of these blanks using the formula:

R = 7,8 V.

Comparing this formula with the formula at = To X , we see that y= R, x = V, and the coefficient of proportionality To= 7.8. The formula is the same, only the letters are different.

Using this formula, let's make a table: let the volume of the 1st blank be 8 cubic meters. cm, then its weight is 7.8 8 \u003d 62.4 (g). The volume of the 2nd blank is 27 cubic meters. cm. Its weight is 7.8 27 \u003d 210.6 (g). The table will look like this:

Calculate the numbers missing in this table yourself using the formula R= d V.

§ 133. Other ways of solving problems with directly proportional quantities.

In the previous paragraph, we solved the problem, the condition of which included directly proportional quantities. For this purpose, we previously derived the direct proportionality formula and then applied this formula. Now we will show two other ways to solve similar problems.

Let's make a problem according to the numerical data given in the table of the previous paragraph.

A task. Blank with a volume of 8 cubic meters. cm weighs 62.4 g. How much will a blank with a volume of 64 cubic meters weigh? cm?

Solution. The weight of iron, as you know, is proportional to its volume. If 8 cu. cm weigh 62.4 g, then 1 cu. cm will weigh 8 times less, i.e.

62.4: 8 = 7.8 (g).

A blank with a volume of 64 cubic meters. cm will weigh 64 times more than a blank of 1 cu. cm, i.e.

7.8 64 = 499.2(g).

We solved our problem by reducing to unity. The meaning of this name is justified by the fact that in order to solve it, we had to find the weight of a unit volume in the first question.

2. Method of proportion. Let's solve the same problem using the proportion method.

Since the weight of iron and its volume are directly proportional quantities, the ratio of two values of one quantity (volume) is equal to the ratio of two corresponding values of another quantity (weight), i.e.

(letter R we denoted the unknown weight of the blank). From here:

(G).

The problem is solved by the method of proportions. This means that in order to solve it, a proportion was made up of the numbers included in the condition.

§ 134. Quantities are inversely proportional.

Consider the following problem: “Five masons can lay down the brick walls of a house in 168 days. Determine in how many days 10, 8, 6, etc. masons could do the same work.

If 5 masons put down the walls of a house in 168 days, then (with the same productivity of labor) 10 masons could do it twice as fast, since on average 10 people do twice as much work as 5 people.

Let's make a table according to which it would be possible to monitor the change in the number of working hours and working hours.

For example, to find out how many days it will take 6 workers, you must first calculate how many days it takes one worker (168 5 = 840), and then six workers (840: 6 = 140). Looking at this table, we see that both quantities have taken six different values. Each value of the first magnitude corresponds more definitely; the value of the second value, for example, 10 corresponds to 84, the number 8 - the number 105, etc.

If we consider the values of both values from left to right, we will see that the values of the upper value increase and the values of the lower value decrease. The increase and decrease is subject to the following law: the values of the number of workers increase as many times as the values of the spent working time decrease. Even more simply, this idea can be expressed as follows: the more workers are employed in any business, the less time they need to do a certain job. The two quantities we encountered in this problem are called inversely proportional.

Thus, if two quantities are interconnected so that with an increase (decrease) in the value of one of them several times, the value of the other decreases (increases) by the same amount, then such quantities are called inversely proportional.

There are many such things in life. Let's give examples.

1. If for 150 rubles. you need to buy several kilograms of sweets, then the number of sweets will depend on the price of one kilogram. The higher the price, the less goods can be bought with this money; this can be seen from the table:

With an increase in the price of sweets several times, the number of kilograms of sweets that can be bought for 150 rubles decreases by the same amount. In this case, the two quantities (the weight of the product and its price) are inversely proportional.

2. If the distance between two cities is 1,200 km, then it can be covered at different times depending on the speed of movement. There are different modes of transportation: on foot, on horseback, by bicycle, by boat, in a car, by train, by plane. The lower the speed, the more time it takes to move. This can be seen from the table:

With an increase in speed several times, the time of movement decreases by the same amount. Hence, under given conditions, speed and time are inversely proportional.

§ 135. The property of inversely proportional quantities.

Let's take the second example, which we considered in the previous paragraph. There we were dealing with two quantities - the speed of movement and time. If we consider the values of these quantities from left to right in the table, we will see that the values of the first quantity (speed) increase, and the values of the second (time) decrease, and speed increases by the same factor as time decreases. It is easy to understand that if you write the ratio of any values of one quantity, then it will not be equal to the ratio of the corresponding values of another quantity. Indeed, if we take the ratio of the fourth value of the upper value to the seventh value (40: 80), then it will not be equal to the ratio of the fourth and seventh values of the lower value (30: 15). It can be written like this:

40:80 is not equal to 30:15, or 40:80 =/= 30:15.

But if instead of one of these ratios we take the opposite, then we get equality, i.e., from these ratios it will be possible to make a proportion. For example:

80: 40 = 30: 15,

40: 80 = 15: 30."

Based on the foregoing, we can draw the following conclusion: if two quantities are inversely proportional, then the ratio of two arbitrarily taken values of one quantity is equal to the inverse ratio of the corresponding values of the other quantity.

§ 136. Inverse proportionality formula.

Consider the problem: “There are 6 pieces of silk fabric of different sizes and different grades. All pieces are the same price. In one piece 100 m of fabric at a price of 20 rubles. per metre. How many meters are in each of the remaining five pieces, if a meter of fabric in these pieces costs 25, 40, 50, 80, 100 rubles, respectively? Let's create a table to solve this problem:

We need to fill in the empty cells in the top row of this table. Let's first try to determine how many meters are in the second piece. This can be done in the following way. It is known from the condition of the problem that the cost of all pieces is the same. The cost of the first piece is easy to determine: it has 100 m and each meter costs 20 rubles, which means that in the first piece of silk for 2,000 rubles. Since the second piece of silk contains the same number of rubles, then, dividing 2,000 rubles. at the price of one meter, that is, at 25, we find the value of the second piece: 2,000: 25 = 80 (m). In the same way, we will find the size of all other pieces. The table will look like:

It is easy to see that there is an inverse relationship between the number of meters and the price.

If you do the necessary calculations yourself, you will notice that each time you have to divide the number 2,000 by the price of 1 m. Conversely, if you now start multiplying the size of a piece in meters by the price of 1 m, you will always get the number 2,000. and it was to be expected, since each piece costs 2,000 rubles.

From this we can draw the following conclusion: for a given pair of inversely proportional quantities, the product of any value of one quantity by the corresponding value of another quantity is a constant number (that is, not changing).

In our problem, this product is equal to 2,000. Check that in the previous problem, which talked about the speed of movement and the time required to move from one city to another, there was also a constant number for that problem (1,200).

Taking into account all that has been said, it is easy to derive the inverse proportionality formula. Denote some value of one quantity by the letter X , and the corresponding value of another value - the letter at . Then, on the basis of the above work X on the at must be equal to some constant value, which we denote by the letter To, i.e.

x y = To.

In this equality X - multiplier, at - multiplier and K- work. By the property of multiplication, the multiplier is equal to the product divided by the multiplicand. Means,

This is the inverse proportionality formula. Using it, we can calculate any number of values of one of the inversely proportional quantities, knowing the values \u200b\u200bof the other and a constant number To.

Consider another problem: “The author of one essay calculated that if his book was in the usual format, then it would have 96 pages, but if it was a pocket format, then it would have 300 pages. He tried different options, started with 96 pages, and then he got 2,500 letters per page. Then he took the number of pages indicated in the table below, and again calculated how many letters would be on the page.

Let's try and calculate how many letters there will be on a page if the book has 100 pages.

There are 240,000 letters in the whole book, since 2,500 96 = 240,000.

Taking this into account, we use the inverse proportionality formula ( at - number of letters per page X - number of pages):

In our example To= 240,000, therefore,

So, there are 2,400 letters on a page.

Similarly, we learn that if the book has 120 pages, then the number of letters on the page will be:

Our table will look like:

Fill in the rest of the cells yourself.

§ 137. Other ways of solving problems with inversely proportional quantities.

In the previous paragraph, we solved problems that included inversely proportional quantities. We previously derived the inverse proportionality formula and then applied this formula. Now we will show two other ways of solving such problems.

1. Method of reduction to unity.

A task. 5 turners can do some work in 16 days. In how many days can 8 turners complete this work?

Solution. There is an inverse relationship between the number of turners and working time. If 5 turners do the work in 16 days, then one person will need 5 times more time for this, i.e.

5 turners do the work in 16 days,

1 turner will complete it in 16 5 = 80 days.

The problem asks, in how many days will 8 turners complete the work. Obviously, they will do the job 8 times faster than 1 turner, i.e. for

80: 8 = 10 (days).

This is the solution of the problem by the method of reduction to unity. Here, first of all, it was necessary to determine the time for the performance of work by one worker.

2. Method of proportion. Let's solve the same problem in the second way.

Since there is an inverse relationship between the number of workers and working time, we can write: the duration of the work of 5 turners the new number of turners (8) the duration of the work of 8 turners the former number of turners (5) Let us denote the desired duration of work by the letter X and substitute in the proportion expressed in words the necessary numbers:

The same problem is solved by the method of proportions. To solve it, we had to make a proportion of the numbers included in the condition of the problem.

Note. In the previous paragraphs, we considered the question of direct and inverse proportionality. Nature and life give us many examples of direct and inverse proportions of quantities. However, it should be noted that these two types of dependence are only the simplest. Along with them, there are other, more complex relationships between quantities. In addition, one should not think that if any two quantities increase simultaneously, then there is necessarily a direct proportionality between them. This is far from true. For example, railroad fares increase with distance: the farther we travel, the more we pay, but this does not mean that the fare is proportional to the distance.

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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ADMINISTRATION OF THE MUNICIPAL FORMATION "CITY OF SARATOV"

MUNICIPAL EDUCATIONAL INSTITUTION

"SEVERAGE SCHOOL № 95 WITH IN-DEPTH

STUDYING INDIVIDUAL SUBJECTS"

Methodical development

algebra lesson in 7th grade

on this topic:

"Direct proportionality

and her schedule.

Mathematic teacher

1 qualification category

Goryunova E.V.

2014 – 2015 academic year

Explanatory note

to the lesson on the topic:

"Direct proportionality and its graph".

Mathematics teacher Goryunova Elena Viktorovna.

Your attention is presented to the lesson in the 7th grade. The teacher works according to the program compiled on the basis of the Model programs of basic general education and the author's program for educational institutions Yu.N. Makarychev. Algebra.7-9 classes // Collection of programs on algebra 7-9 classes. M. Enlightenment, 2009 compiled by T.A. Burmistrov. The program corresponds to the algebra textbook Yu.N. Makarychev, N.G. Mindyuk, K.I. Neshkov., S.B. Suvorov., edited by S.A. Telyakovsky "Algebra Grade 7" (publishing house "Enlightenment" 2009).

14 hours are allotted for studying the topic "Functions", of which 6 hours are for the section "Functions and their graphs", 3 hours - for the section "Direct proportionality and its graph", 4 hours for the section "Linear function and its graph" and 1 hour K/R.

GOALS:

Educational:

Developing:

3. Encourage students to self-control and mutual control.

Educational:

To instill a sense of respect for classmates, attention to the word, to promote the education of independence, responsibility, accuracy in the construction of drawings

These goals are achieved through a number of tasks:

Lesson equipment:

The lesson used individual cards with tasks and a multimedia projector, all the facts about R. Descartes were taken by the teacher on the Internet from official media sites and redesigned specifically for this lesson, taking into account the topic of the lesson, the textbook.

Lesson type and structure:

This lesson is a lesson in mastering new knowledge and skills (types of lessons according to V.A. Onischuk), so it was rational to apply elements of research activity.

Implementation of learning principles:

The following principles were implemented in the lesson:

Scientific teaching.

The principle of systematic and consistent teaching was carried out with constant reliance on previously studied material.

Consciousness, activity and independence of students was achieved in the form of stimulating cognitive activity with the help of effective techniques and visual aids (such as slide shows, providing historical facts and information from the life of the mathematician and philosopher R. Descartes, individual printed sheets of students.

The lesson was implemented the principle of comfort.

Forms and methods of teaching:

During the lesson, various forms of training were used - this is individual and frontal work, mutual verification. Such forms are more rational for this type of lesson, as they allow the child to develop independent thinking, critical thinking, the ability to defend their point of view, the ability to compare and draw conclusions.

The main method of this lesson is the partial search method, which is characterized by the work of students in solving problematic cognitive tasks.

Phys. a minute was both physical exercise and consolidation of the material just studied.

At the end of the lesson, it is advisable to summarize the work done in the lesson.

General results of the lesson:

I believe that the tasks set for the lesson were implemented, the children applied their knowledge in a new situation, everyone could express their point of view. The use of visualization in the form of a presentation, individual printed sheets of students allows you to motivate students at each stage of the lesson and avoid overloading and overworking students.

Lesson topic:

Didactic task: familiarity with direct proportionality and the construction of its graph.

Goals:

Educational:

1. Organize the activities of students on the perception of the topic "Direct proportionality and its schedule" and primary consolidation: determining direct proportionality and plotting its schedule, to form skills for competent plotting

2. To create conditions for creating a system of basic knowledge and skills in the memory of students, to stimulate search activity

Developing:

1. To develop analytical and synthesizing thinking (to promote the development of observation, the ability to analyze, the development of skills to classify facts, to draw generalizing conclusions).

2. Develop abstract thinking (development of skills to identify common and essential features, to distinguish non-essential features and to be distracted from them).

3. Encourage students to self-control and mutual control

Educational:

To instill a sense of respect for classmates, attention to the word, to promote the education of independence, responsibility, accuracy in the construction of drawings.

Equipment: computer, presentation, printed cards with tasks for each student.

Lesson plan:

1. Organizational moment.

2.Motivation of the lesson.

3.Updating knowledge.

4. Study of new material.

5. Fixing the material.

6. The result of the lesson.

During the classes.

1. Organizational moment.

Good morning, guys! I would like to start the lesson with the following words. (Slide 1)

The French scientist René Descartes once remarked: “I think, therefore I am.”

The guys prepared a message about the French scientist R. Descartes.

Rene Descartes is better known as a great philosopher than a mathematician. But it was he who was the pioneer of modern mathematics, and his merits in this area are so great that he is justly included among the great mathematicians of our time.

Student message:(Slide 2)

Born Descartes was born in France, in the small town of Lae. His father was a lawyer, his mother died when Rene was 1 year old. After graduating from a college for the sons of aristocratic families, he, following the example of his brother, began to study law. At the age of 22, he left France and, as a volunteer officer, served in the troops of various military leaders who participated in the 13-year war. Descartes in his philosophical doctrine developed the idea of the omnipotence of the human mind, and therefore was persecuted by the Catholic Church. Wanting to find a safe haven for quiet work in philosophy and mathematics, which he had been interested in since childhood, Descartes settled in Holland in 1629, where he lived almost to the end of his life. All major works of Descartes on philosophy, mathematics, physics, cosmology and physiology were written by him in Holland.

Mathematical works of Descartes are collected in his book "Geometry" (1637). In "Geometry" Descartes gave the foundations of analytic geometry and algebra. Descartes was the first to introduce the concept of a variable function into mathematics. He drew attention to the fact that a curve on a plane is characterized by an equation that has the property that the coordinates of any point lying on this line satisfy this equation. He divided the curves given by an algebraic equation into classes depending on the largest power of the unknown quantity in the equation. Descartes introduced into mathematics the plus and minus signs to denote positive and negative quantities, the notation of the degree, and the sign to denote an infinitely large quantity. For variables and unknown quantities, Descartes adopted the designations x, y, z, and for known and constant quantities -a .b .c, as you know, these designations are used in mathematics to this day. Despite the fact that Descartes did not advance very far in the field of analytic geometry, his works had a decisive influence on the further development of mathematics. For 150 years, mathematics has developed along the lines outlined by Descartes.

Let's follow the advice of the scientist. We will be active, attentive, we will reason, think and learn new things, because knowledge will be useful to you in later life. And I would like to offer these words (Slide 3) of R. Descartes as the motto of our lesson: "Respect for others gives rise to respect for oneself."

2.Motivation.

Let's check with what mood you came to the lesson. We draw a smiley on the margins.

Take cards. The words of R. Descartes are also written here: “ In order to improve your mind, you need to reason more than memorize. These words will guide us in our work.

Task number 1 with mathematical terms that we will use in the lesson. Correct the mistakes made in the spelling of these terms. (Slide 4)

Change leaflets and check if all the errors are corrected. (Slide 5) What did you notice? Which word has no errors? (function, graph)

3. Actualization of knowledge.

a) We got acquainted with the concept of "function" in the previous lessons. Let's recall the basic concepts and definitions on this topic.

We also worked with function graphs. Which of the words of the dictation did we use when working on the topic "Graphs of functions"? What do they stand for?

On this slide, determine which of the lines will be the graph of the function? (Slide 6)

And who will say what we will talk about in this lesson? What are the goals for the lesson? (slide 7)

On the sheets of students write the number and write the topic of the lesson: "Direct proportionality and its graph"

Recall the material of previous lessons

Write formulas to solve the following problems. (Slide 9,10)

What are the dependent and independent variables? What depends on what? What addiction? (Slide)

Which of the formulas is different from the others? (Slide)

c) How can formulas be written in general form? (Slide)

y =kx , y - dependent variable

x - independent variable

k - constant number (coefficient)

We wrote down the formula, and this is one way to define a function. Direct proportional dependence is a function.

4. Study of new material.

Definition. Direct proportionality is a function that can be specified by the formula y \u003d kx, where x is an independent variable, and k is a certain number that is not equal to zero, the coefficient of direct proportionality (a constant ratio of proportional values)

Read the rule in the textbook on page 65

The scope of this function? (The set of all numbers)

Fixing the material.

Complete the task in sheets No. 4 (Slide) Divide the formulas into 2 groups according to the topic of the lesson: (read the rule in the textbook on p. 65)

y=2x, y=3x-7, y=-0.2x, y=x, y=x², y=x, y=-5.8+3x, y=-x, y=50x,

1st group: _____________________________________________________

2nd group: _____________________________________________________

Underline the direct proportionality factor.

We carry out No. 298 on page 68 (orally), I dictate, you determine the formula of proportionality by ear and screw your eyes, if not proportionality, then rotate your eyes from left to right.

Come up with and write down 4 formulas for the direct proportionality function:

1)y=_________2)y=__________3)y=_________4)y=__________

Learning new material

What is the graph of this function? Do you want to know?

We have already built a function graph in task No. 2, can we call this function pr. proportionality? So we have already built a graph of pr.proportionality. Rule in the textbook on page 67.

Let's see how we will build a graph of this function (Slide)

Fixing the material.

Let's build a graph number 7 in the sheets of students (Slide)

What point will we have in any graph of proportionality?

We work according to ready-made drawings. (Slide)

Conclusion: the graph is a straight line passing through the origin.

T.K. the graph is a straight line, how many points are needed to plot it? One already exists (0;0)

We carry out No. 300

Summary of the lesson. Let's summarize the work in today's lesson (Slide). They did everything. What have you planned?

Reflection. (Slide)

Check the mood of the students at the end of the lesson. (Smiley) (Slide)

>>Math: Direct proportionality and its graph

Direct proportionality and its graph

Among the linear functions y = kx + m, the case when m = 0 is highlighted; in this case takes the form y = kx and it is called direct proportionality. This name is explained by the fact that two quantities y and x are called directly proportional if their ratio is equal to a specific
a number other than zero. Here , this number k is called the coefficient of proportionality.

Many real situations are modeled using direct proportionality.

For example, the path s and time t at a constant speed, 20 km/h, are related by the dependence s = 20t; this is a direct proportionality, with k = 20.

Another example:

the cost y and the number x of loaves of bread at the price of 5 rubles. per loaf are linked by the dependence y = 5x; this is a direct proportionality, where k = 5.

Proof. Let's do it in two stages.
1. y \u003d kx is a special case of a linear function, and the graph of a linear function is a straight line; let's denote it by I.
2. The pair x \u003d 0, y \u003d 0 satisfies the equation y - kx, and therefore the point (0; 0) belongs to the graph of the equation y \u003d kx, that is, the line I.

Therefore, the line I passes through the origin. The theorem has been proven.

One must be able to move not only from the analytical model y \u003d kx to the geometric one (direct proportionality graph), but also from the geometric models to analytical. Consider, for example, a straight line on the xOy coordinate plane shown in Figure 50. It is a direct proportionality graph, you just need to find the value of the coefficient k. Since y, it is enough to take any point on the line and find the ratio of the ordinate of this point to its abscissa. The straight line passes through the point P (3; 6), and for this point we have: Hence, k = 2, and therefore the given straight line serves as a graph of direct proportionality y \u003d 2x.

As a result, the coefficient k in the notation of the linear function y \u003d kx + m is also called the slope. If k>0, then the line y \u003d kx + m forms an acute angle with the positive direction of the x axis (Fig. 49, a), and if k< О, - тупой угол (рис. 49, б).

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Direct and inverse proportionality

If t is the walking time (in hours), s is the distance traveled (in kilometers), and he moves uniformly at a speed of 4 km/h, then the relationship between these quantities can be expressed by the formula s = 4t. Since each value of t corresponds to a unique value of s, we can say that a function is given using the formula s = 4t. It is called direct proportionality and is defined as follows.

Definition. Direct proportionality is a function that can be specified using the formula y \u003d kx, where k is a non-zero real number.

The name of the function y \u003d k x is due to the fact that in the formula y \u003d kx there are variables x and y, which can be values of quantities. And if the ratio of two values \u200b\u200bis equal to some number other than zero, they are called directly proportional . In our case = k (k≠0). This number is called proportionality factor.

The function y \u003d k x is a mathematical model of many real situations considered already in the initial course of mathematics. One of them is described above. Another example: if there are 2 kg of flour in one package, and x such packages are bought, then the entire mass of the purchased flour (we denote it by y) can be represented as a formula y \u003d 2x, i.e. the relationship between the number of packages and the total mass of purchased flour is directly proportional with the coefficient k=2.

Recall some properties of direct proportionality, which are studied in the school course of mathematics.

1. The domain of the function y \u003d k x and the domain of its values is the set of real numbers.

2. The graph of direct proportionality is a straight line passing through the origin. Therefore, to construct a graph of direct proportionality, it is enough to find only one point that belongs to it and does not coincide with the origin, and then draw a straight line through this point and the origin.

For example, to plot the function y = 2x, it is enough to have a point with coordinates (1, 2), and then draw a straight line through it and the origin (Fig. 7).

3. For k > 0, the function y = kx increases over the entire domain of definition; for k< 0 - убывает на всей области определения.

4. If the function f is a direct proportionality and (x 1, y 1), (x 2, y 2) - pairs of corresponding values of the variables x and y, and x 2 ≠ 0 then.

Indeed, if the function f is a direct proportionality, then it can be given by the formula y \u003d kx, and then y 1 \u003d kx 1, y 2 \u003d kx 2. Since at x 2 ≠0 and k≠0, then y 2 ≠0. That's why and means .

If the values of the variables x and y are positive real numbers, then the proved property of direct proportionality can be formulated as follows: with an increase (decrease) in the value of the variable x several times, the corresponding value of the variable y increases (decreases) by the same amount.

This property is inherent only in direct proportionality, and it can be used in solving word problems in which directly proportional quantities are considered.

Task 1. In 8 hours, the turner made 16 parts. How many hours will it take a turner to make 48 parts if he works at the same productivity?

Solution. The problem considers the quantities - the turner's working time, the number of parts made by him and productivity (i.e., the number of parts manufactured by the turner in 1 hour), the latter value being constant, and the other two taking different values. In addition, the number of parts made and the time of work are directly proportional, since their ratio is equal to a certain number that is not equal to zero, namely, the number of parts made by the turner in 1 hour. If the number of parts made is denoted by the letter y, the work time is x, and performance - k, then we get that = k or y = kx, i.e. the mathematical model of the situation presented in the problem is direct proportionality.

The problem can be solved in two arithmetic ways:

1 way: 2 way:

1) 16:8 = 2 (children) 1) 48:16 = 3 (times)

2) 48:2 = 24(h) 2) 8-3 = 24(h)

Solving the problem in the first way, we first found the proportionality coefficient k, it is equal to 2, and then, knowing that y \u003d 2x, we found the value of x, provided that y \u003d 48.

When solving the problem in the second way, we used the property of direct proportionality: how many times the number of parts made by a turner increases, the amount of time for their manufacture increases by the same amount.

Let us now turn to the consideration of a function called inverse proportionality.

If t is the time of the pedestrian's movement (in hours), v is his speed (in km/h) and he walked 12 km, then the relationship between these values can be expressed by the formula v∙t = 20 or v = .

Since each value of t (t ≠ 0) corresponds to a single value of velocity v, we can say that a function is given using the formula v = . It is called inverse proportionality and is defined as follows.

Definition. Inverse proportionality is a function that can be specified using the formula y \u003d, where k is a non-zero real number.

The name of this function comes from the fact that y= there are variables x and y, which can be values of quantities. And if the product of two quantities is equal to some number other than zero, then they are called inversely proportional. In our case, xy = k(k ≠ 0). This number k is called the coefficient of proportionality.

Function y= is a mathematical model of many real situations considered already in the initial course of mathematics. One of them is described before the definition of inverse proportionality. Another example: if you bought 12 kg of flour and put it in l: jars of y kg each, then the relationship between these quantities can be represented as x-y \u003d 12, i.e. it is inversely proportional with the coefficient k=12.

Recall some properties of inverse proportionality, known from the school course of mathematics.

1. Function scope y= and its range x is the set of non-zero real numbers.

2. The inverse proportionality graph is a hyperbola.

3. For k > 0, the branches of the hyperbola are located in the 1st and 3rd quadrants and the function y= is decreasing on the entire domain of x (Fig. 8).

Rice. 8 Fig.9

When k< 0 ветви гиперболы расположены во 2-й и 4-й четвертях и функция y= is increasing over the entire domain of x (Fig. 9).

4. If the function f is inversely proportional and (x 1, y 1), (x 2, y 2) are pairs of corresponding values of the variables x and y, then.

Indeed, if the function f is inversely proportional, then it can be given by the formula y= ,and then . Since x 1 ≠0, x 2 ≠0, x 3 ≠0, then

If the values of the variables x and y are positive real numbers, then this property of inverse proportionality can be formulated as follows: with an increase (decrease) in the value of the variable x several times, the corresponding value of the variable y decreases (increases) by the same amount.

This property is inherent only in inverse proportionality, and it can be used in solving word problems in which inversely proportional quantities are considered.

Problem 2. A cyclist, moving at a speed of 10 km/h, covered the distance from A to B in 6 hours.

Solution. The problem considers the following quantities: the speed of the cyclist, the time of movement and the distance from A to B, the latter value being constant, and the other two taking different values. In addition, the speed and time of movement are inversely proportional, since their product is equal to a certain number, namely the distance traveled. If the time of the cyclist's movement is denoted by the letter y, the speed is x, and the distance AB is k, then we get that xy \u003d k or y \u003d, i.e. the mathematical model of the situation presented in the problem is inverse proportionality.

You can solve the problem in two ways:

1 way: 2 way:

1) 10-6 = 60 (km) 1) 20:10 = 2 (times)

2) 60:20 = 3(4) 2) 6:2 = 3(h)

Solving the problem in the first way, we first found the proportionality coefficient k, it is equal to 60, and then, knowing that y \u003d, we found the value of y, provided that x \u003d 20.

When solving the problem in the second way, we used the inverse proportionality property: how many times the speed of movement increases, the time to travel the same distance decreases by the same amount.

Note that when solving specific problems with inversely proportional or directly proportional quantities, some restrictions are imposed on x and y, in particular, they can be considered not on the entire set of real numbers, but on its subsets.

Problem 3. Lena bought x pencils, and Katya bought 2 times more. Denote the number of pencils Katya bought as y, express y in terms of x, and plot the established correspondence graph, provided that x ≤ 5. Is this match a function? What is its domain of definition and range of values?

Solution. Katya bought u = 2 pencils. When plotting the function y=2x, it must be taken into account that the variable x denotes the number of pencils and x≤5, which means that it can only take on the values 0, 1, 2, 3, 4, 5. This will be the domain of this function. To get the range of this function, you need to multiply each value x from the domain of definition by 2, i.e. it will be a set (0, 2, 4, 6, 8, 10). Therefore, the graph of the function y \u003d 2x with the domain of definition (0, 1, 2, 3, 4, 5) will be the set of points shown in Figure 10. All these points belong to the line y \u003d 2x.