Direct proportional graph

Lesson Objectives:

Determine the type of graph of direct proportionality;

Investigate the dependence of the location of the direct proportionality graph on coordinate plane from the sign of the number k;

To form the ability to build a graph of direct proportionality according to the formula and perform the reverse action - write down the formula of the function according to the graph;

Contribute to the education of independence, responsibility, accuracy in the construction of drawings;

Learn to pose and solve problems;

To cultivate the will and perseverance to achieve the final results, respect for classmates.

Planned results:

Subject skills: repetition of theoretical material on a given topic; the formation of knowledge and skills on the material being studied, the consolidation of skills in constructing a graph of direct proportionality;

Personal UUD: the formation of skills of introspection and self-control, the skills of compiling algorithms to complete the task, sustainable motivation for learning;

Regulatory UUD: defining the goal, searching for means to achieve it, identifying deviations from the standard in their work, understanding the causes of errors;

Cognitive UUD: the ability to replace terms with definitions, highlighting and formulating a problem, expressing the meaning of a situation using an algorithm;

Communicative UUD: regulation of one's own activity through speech actions, the ability to organize educational interaction in a team, a couple, the ability to express a point of view, substantiating it with reason.

Corrective component of the lesson:

Multiple repetition of information using materialized supports;

Drawing up and application of the algorithm;

Automation of pronunciation and writing of terms with a complex syllabic structure.

Lesson type: mastering new knowledge and skills using elements.

Learning principles:

scientific;

Consistency and consistency;

visibility;

Comfort.

Teaching methods: individual, frontal, group, verbal-visual, partially search.

Technical support of the lesson: computer, projector, multimedia presentation.

Equipment: a portrait of R. Descartes, a poster with a statement, drawing tools, colored pencils, cards for individual and collective work of students; Handout.

Textbook: “Algebra. Grade 7 ": a textbook for educational institutions / [,]; ed. . – 19th ed. – M.: Enlightenment, 2012.

Lesson plan:

1. Organizational moment.

2. Lesson motivation.

3. Update basic knowledge students.

4. Formulation of the topic of the lesson, goals, objectives.

5. The main stage of the lesson:

1) mastering new knowledge by following the instructions;

2) drawing up an algorithm for constructing a graph of direct proportionality;

3) research work.

6. Physical education.

7. Primary fastening:

1) fulfillment of tasks for working out the algorithm;

2) independent work.

8. Homework.

9. The result of the lesson.

10. Reflection.

During the classes

I. Organizational moment.

(Slide 1) Mutual greeting. Check readiness for the lesson.

II. Motivation.

1. (Slide 2) - I would like to start the lesson with the following words: “I think, therefore I am”, which were said by the French scientist Rene Descartes.

Rene Descartes is better known as a great philosopher. But it is precisely in mathematics that his merits are so great that he is justly included among the great mathematicians. The guys prepared messages about the life and work of Descartes.

(Slide 3) Message 1. 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 began to study, following the example of his brother. At the age of 22, he left France and served as a volunteer officer in various troops.

Descartes in his philosophical teaching 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 was 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.

(Slide 4) Message 2. Descartes introduced into mathematics the signs "+" and "-" to denote positive and negative quantities, the notation of the degree and the sign to denote an infinitely large value. For variables and unknown quantities, Descartes adopted the designation x, y, z, and for known and constant quantities, a, b, c. These notations are used in mathematics to this day. He introduced the coordinate system that was named after him. 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 of R. Descartes as the motto of our lesson: "Respect for others gives rise to respect for oneself."

2. - And now let's work with mathematical terms which we will use in the lesson. Complete task number 1 from the card on your own.

Card, task 1. Correct the mistakes made in the spelling of terms:

coordinate

Ardinata

Coefficient

argument

variable

Swap cards and check if all the errors are corrected.

(Slide 5) - Let's check the slide.

III. Knowledge update.

- Let's recall the main material of the previous lessons, on which we will rely.

1. Define direct proportionality.

2. (Slide 6) - Determine by the formula which of the functions is directly proportional:

a) y = 182x; c) y \u003d -17x2;

b) y = ; d) y \u003d 3x + 11.

3. Card, task 2. Divide the formulas into 2 groups. In the first group, write down the functions that are direct proportionality, in the second - those that are not. For direct proportions, underline the coefficient k.

y = 2x; y \u003d 3x - 7; y \u003d -0.2x; y = ; y = x2; y = x; y = 8 + 3x; y = - x; y = 70x

(Slide 7) - Check yourself. Who completed without errors? Well done. I see that you have prepared well for the lesson and are ready to learn new material.

IV. Formulation of the topic of the lesson, goals, objectives.

Now we have considered the direct proportionality given by the formula. Think about how else you can set this function? Which method is more visual? So, the topic of our lesson is ... (students formulate).

Students write the topic of the lesson in a notebook.

On the leading questions of the teacher, students formulate the goals and objectives of the lesson.

V. The main stage of the lesson.

1. - Let's do a little practical work.

Each student receives a piece of paper with a direct proportionality formula. The goal is to work with the formula according to the instructions recorded in task 3 cards.

(Slide 8) y \u003d x y \u003d - x

y = 1.5x y = -1.5x

Card, task 3. Instruction:

fill in the table of function values ​​at -3 ≤ x ≤ 3 with step 1; mark in the coordinate plane the points whose coordinates are placed in the table; connect the dots.

The students then answer the teacher's questions:

How are the points you plotted located?

What happens when you connect the dots?

What is the peculiarity in the location of a straight line in the coordinate plane?

What conclusion can be drawn from this?

Students formulate a conclusion about the form of a direct proportionality graph and its features.

Let's find it in the textbook and compare it with the one we got.

2. - To build a straight line, how many points do we need to know?

We already have one. Which?

So, how many points do we still need to have to plot a direct proportionality graph?

Based on these conclusions, students draw up an algorithm for constructing a graph of direct proportionality.

Algorithm

1. Find the coordinates of some point of the graph of this function (other than the origin).

2. Mark this point on the coordinate plane.

3. Draw a line through this point and the origin.

3. - And now we will conduct a small study and draw a conclusion, and which one - you will find out later.

Raise your hands those who had a function with a positive coefficient k. In what coordinate quarters are your graphs located?

Raise your hands those who had a function with a negative coefficient k. In what coordinate quarters are your graphs located?

As a result research work students draw a conclusion about the location of direct proportionality graphs depending on the sign of the coefficient k and compare with the conclusions in the textbook.

VI. Fizkultminutka. (Slide 10)

Get up quickly and smile

Pulled higher and higher.

Come on, straighten your shoulders

Raise, lower.

Turn right, turn left

Touch your hands with your knees.

Sit down, get up. Sit down, get up.

And they ran on the spot.

VII. Primary fastening.

1. Performing a task to work out the algorithm for constructing a graph of direct proportionality, finding the values ​​of a function according to the graph using a known value of the argument and vice versa.

Students complete No. 000 (a, b) from the textbook in notebooks and on the board.

When completing this task, we repeat with the students the rule of finding the value of a function according to the graph for a given value of the argument and vice versa (mark a point on the abscissa axis; draw a straight line perpendicular to the abscissa axis until it intersects with the function graph; from the resulting point we lower the perpendicular to the y-axis and find the corresponding ordinate value).

Also in this example we show that it is very important to choose the correct value of the unit segment and the abscissa of the selected point.

2. Independent work(subject to availability of time).

Work on drawing 26 from the textbook.

(Slide 11) - What do you think, is it possible to write down its analytical formula using the graph of a function?

We find out together with the students that all graphs are straight lines passing through the origin, which means that the functions are direct proportions and they can be specified by a formula of the form y \u003d kx. The problem is reduced to finding the coefficient k. To do this, on each graph, select an arbitrary point with integer coordinates.

(Slide 12) - Check yourself.

VIII. Homework: item 15 (learn the rules); No. 000 (a), 301 (b) - build graphs according to the algorithm; 302 - answer a question, think over a solution.

IX. Summary of the lesson.

What did we work on in class today?

What is a direct proportional graph?

What is the graphing algorithm?

How is the graph of the function y \u003d kx located in the coordinate plane for k< 0 и при k > 0?

X. Reflection. (Slide 14)

Were you interested in the lesson?

Who thinks he worked well today?

What difficulties did you have in class?

(Slide 15) - You did a good job in the lesson. Well done! I would especially like to note ... Thank you all! The lesson is over.

Proportionality- this is the dependence of one quantity on another, in which a change in one quantity leads to a change in the other by the same amount.

The proportionality of values ​​can be direct and inverse.

Direct proportionality

Direct proportionality- this is the dependence of two quantities, in which one quantity depends on the second quantity so that their ratio remains unchanged. Such quantities are called directly proportional or simply proportional.

Consider an example of direct proportionality on the path formula:

s = vt

Where s is the way v- speed, and t- time.

With uniform motion, the distance is proportional to the time of motion. If we take the speed v equal to 5 km/h, then the distance traveled s will depend on the travel time. t:

Speed v= 5 km/h
Time t(h)	1	2	4	8	16
Path s(km)	5	10	20	40	80

It can be seen from the example that how many times the movement time increases t, the distance traveled increases by the same amount s. In the example, we increased the time every time by 2 times, since the speed did not change, then the distance also doubled.

In this case, the speed ( v\u003d 5 km / h) is a direct proportionality coefficient, that is, the ratio of the path to time, which remains unchanged:

If the time of movement remains unchanged, then with uniform movement the distance will be proportional to the speed:

It follows from these examples that Two quantities are said to be directly proportional if, when one of them increases (or decreases) several times, the other increases (or decreases) by the same amount..

Formula of direct proportionality

Formula of direct proportionality:

y = kx

Where y And x k is a constant value called the coefficient of direct proportionality.

Direct proportionality coefficient is the ratio of any corresponding values ​​of the proportional variables y And x equal to the same number.

Direct proportionality formula:

y	= k
x

Inverse proportionality

Inverse proportionality is the relationship between two quantities, in which an increase in one value leads to a proportional decrease in the other. Such quantities are called inversely proportional.

Consider an example of inverse proportionality on the path formula:

s = vt

Where s is the way v- speed, and t- time.

When passing the same path at different speeds, the time will be inversely proportional to the speed. If you take the path s equal to 120 km, then the time spent on overcoming this path t will only depend on the speed v:

Path s= 120 km
Speed v(km/h)	10	20	40	80
Time t(h)	12	6	3	1,5

The example shows that how many times the speed of movement increases v, the time decreases by the same amount t. In the example, we increased the speed of movement by 2 times each time, and since the distance to be overcome did not change, the amount of time to overcome this distance was also halved.

In this case, the path ( s= 120 km) is an inverse proportionality coefficient, that is, the product of speed and time:

s = vt, therefore 10 12 = 20 6 = 40 3 = 80 1.5 = 120

From this example it follows that two quantities are said to be inversely proportional if when one of them increases several times, the other decreases by the same amount.

Inverse proportion formula

Inverse proportion formula:

y =	k
	x

Where y And x- This variables, A k is a constant value called the inverse proportionality coefficient.

Inverse proportionality factor is the product of any corresponding values ​​of the inversely proportional variables y And x equal to the same number.

The formula for the inverse proportionality coefficient.

Let's assume that t is the time of the pedestrian's movement (in seconds), s is the distance traveled by him (in meters). If the pedestrian moves uniformly at a speed of 5 m/s, then s = 5t. It is logical that each value of the variable t corresponds to a single value s. The formula s = 5t, where t ≥ 0, defines a function.

Suppose that n is the number of ice cream packs, p is their cost (in rubles). If the price of one pack of ice cream is 6 rubles, then p = 6n. It is logical that each value of the variable n corresponds to a single value p.

The formula p = 6n, where n € N, defines a function.

In the considered examples, we worked with functions given by formulas of the form y \u003d kx, where x and y are variables, k is a non-zero number.

A function that can be specified by a formula of the form y \u003d kx, where k is a non-zero number, is called direct proportionality (= proportionality).

The number k is called the coefficient of proportionality. The variable y is said to be proportional to the variable x.

The domain of definition of direct proportionality can be the set of all numbers or some of its subsets. In the given examples, in the first case, the function was defined on the set of positive numbers, in the second case, on the set of natural numbers.

From the formula y \u003d kx for x ≠ 0, it follows that y / x \u003d k. The converse is also true: if y/x = k, then y = kx. Therefore, in order to find out whether the function x - y is directly proportional, the quotients y / x are compared for all pairs of corresponding values ​​​​of the variables x and y, in which x ≠ 0. If these quotients are equal to the same non-zero number k, and if x equal to 0 corresponds to y equal to 0 (if 0 is in the domain of the function), then the dependence of y on x is a direct proportionality.

Consider the theory in practice and analyze the example.

Example. The function a – b is given by the values

If a = -4, then b = -12. If a = -3, then b = -9. If a = -1.5, then b = -4.5. If a = 2.5, then b = 7.5. If a = 5, then b = 15. If a = 6.1, then b = 18.3.

Is this function directly proportional?

For each pair (a; b) of the corresponding values ​​of the variables a and b, we find the quotient b/a.

If a = -4, then b = -12, then k = 3. If a = -3, then b = -9, then k = 3. If a = -1.5, then b = -4, 5, so k = 3. If a = 2.5, then b = 7.5, then k = 3. If a = 5, then b = 15, then k = 3. If a = 6.1 , then b = 18.3, hence k = 3.

It turns out that the found quotients are equal to the same number 3. Hence, the function f we are considering is a direct proportionality.

Direct proportionality is characterized by certain properties.

If the function x - y is a direct proportionality and (x 1; y 1), (x 2; y 2) are pairs of corresponding values ​​​​of the variables x and y, and x 2 ≠ 0, then x 1 / x 2 = y 1 / y 2.

Proof.

Let k be the coefficient of proportionality. From the formula y \u003d kx we have that y 1 \u003d kx 1, y 2 \u003d kx 2 (because x 2 ≠ 0 and k ≠ 0, then y 2 ≠ 0). From here we get y 1 / y 2 \u003d kx 1 / kx 2 \u003d x 1 / x 2.

If the values ​​of the variables x and y are positive numbers, then we can formulate the proven property of direct proportionality as follows:

with an increase in the value of x several times, the corresponding value of y increases by the same amount; similarly: with a decrease in the value of x by several times, the corresponding value of y increases by the same amount.

The established property of direct proportionality is convenient to use when solving problems.

In 8 hours, the turner made 17 parts. How many hours will it take a turner to make 85 parts if he works at the same productivity?

Solution.

Let the turner need x hours to make 85 parts. at constant productivity, the number of parts manufactured is directly proportional to the time spent, then 8/x \u003d 17/85.

Hence 17x = 8 ∙ 85; x \u003d (8 ∙ 85) / 17; x = 40.

Answer: the turner will need 40 hours.

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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 dependency, at 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 functional dependency, at which an increase in the independent value (argument) causes a proportional decrease in the dependent value (function).

Mathematically, inverse proportionality is written as a formula:

Function properties:

Sources

Wikimedia Foundation. 2010 .

• Newton's second law
• Coulomb barrier

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