Direct proportionality and its graph - Knowledge Hypermarket. Direct proportional dependence

I. Directly proportional quantities.

Let the value y depends on the size X. If when increasing X several times the size at increases by the same amount, then such values X And at are called directly proportional.

Examples.

1 . The quantity of goods purchased and the purchase price (with a fixed price for one unit of goods - 1 piece or 1 kg, etc.) How many times more goods were bought, the more times more they paid.

2 . The distance traveled and the time spent on it (at constant speed). How many times longer is the path, how many times more time will it take to complete it.

3 . The volume of a body and its mass. ( If one watermelon is 2 times larger than another, then its mass will be 2 times larger)

II. Property of direct proportionality of quantities.

If two quantities are directly proportional, then the ratio of two arbitrarily taken values of the first quantity is equal to the ratio of two corresponding values of the second quantity.

Task 1. For raspberry jam we took 12 kg raspberries and 8 kg Sahara. How much sugar will you need if you took it? 9 kg raspberries?

Solution.

We reason like this: let it be necessary x kg sugar for 9 kg raspberries The mass of raspberries and the mass of sugar are directly proportional quantities: how many times less raspberries are, the same number of times less sugar is needed. Therefore, the ratio of raspberries taken (by weight) ( 12:9 ) will be equal to the ratio of sugar taken ( 8:x). We get the proportion:

12: 9=8: X;

x=9 · 8: 12;

x=6. Answer: on 9 kg raspberries need to be taken 6 kg Sahara.

The solution of the problem It could be done like this:

Let on 9 kg raspberries need to be taken x kg Sahara.

(The arrows in the figure are directed in one direction, and up or down does not matter. Meaning: how many times the number 12 more number 9 , the same number of times 8 more number X, i.e. there is a direct relationship here).

Answer: on 9 kg I need to take some raspberries 6 kg Sahara.

Task 2. Car for 3 hours traveled the distance 264 km. How long will it take him to travel? 440 km, if he drives at the same speed?

Solution.

Let for x hours the car will cover the distance 440 km.

Answer: the car will pass 440 km in 5 hours.

Linear function

Linear function is a function that can be specified by the formula y = kx + b,

where x is the independent variable, k and b are some numbers.

The graph of a linear function is a straight line.

The number k is called slope of a straight line– graph of the function y = kx + b.

If k > 0, then the angle of inclination of the straight line y = kx + b to the axis X spicy; if k< 0, то этот угол тупой.

If the slopes of the lines that are graphs of two linear functions are different, then these lines intersect. And if the angular coefficients are the same, then the lines are parallel.

Graph of a function y =kx +b, where k ≠ 0, is a line parallel to the line y = kx.

Direct proportionality.

Direct proportionality is a function that can be specified by the formula y = kx, where x is an independent variable, k is a non-zero number. The number k is called coefficient of direct proportionality.

The graph of direct proportionality is a straight line passing through the origin of coordinates (see figure).

Direct proportionality is a special case of a linear function.

Function Propertiesy =kx:

Inverse proportionality

Inverse proportionality is called a function that can be specified by the formula:

k
y = -
x

Where x is the independent variable, and k– a non-zero number.

The graph of inverse proportionality is a curve called hyperbole(see picture).

For a curve that is the graph of this function, the axis x And y act as asymptotes. Asymptote- this is the straight line to which the points of the curve approach as they move away to infinity.

k
Function Propertiesy = -:
x

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 decimal fractions;
develop students' logical thinking.

DURING THE CLASSES

I. Self-determination for activity(Organizing time)

- Guys! Today in the lesson we will get acquainted with problems solved using proportions.

II. Updating knowledge and recording difficulties in activities

2.1. Oral work (3 min)

– Find the meaning of the expressions and find out the word encrypted in the answers.

14 – s; 0.1 – and; 7 – l; 0.2 – a; 17 – in; 25 – to

– The resulting word is strength. Well done!
– The motto of our lesson today: Power is in knowledge! I'm searching - that means I'm learning!
– Make up a proportion from the resulting numbers. (14:7 = 0.2:0.1 etc.)

2.2. Let's consider the relationship between the quantities we know (7 min)

– the distance covered by the car at a constant speed, and the time of its movement: S = v t ( with increasing speed (time), the distance increases);
– vehicle speed and time spent on the journey: v=S:t(as the time to travel the path increases, the speed decreases);
– the cost of goods purchased at one price and its quantity: C = a · n (with an increase (decrease) in price, the purchase cost increases (decreases));
– price of the product and its quantity: a = C: n (with an increase in quantity, the price decreases)
– area of the rectangle and its length (width): S = a · b (with increasing length (width), the area increases;
– rectangle length and width: a = S: b (as the length increases, the width decreases;
– the number of workers performing some work with the same labor productivity, and the time it takes to complete this work: t = A: n (with an increase in the number of workers, the time spent on performing the work decreases), etc.

We have obtained dependences in which, with an increase in one quantity several times, another immediately increases by the same amount (examples are shown with arrows) and dependences in which, with an increase in one quantity several times, the second quantity decreases by the same number of times.
Such dependencies are called direct and inverse proportionality.
Directly proportional dependence– a relationship in which as one value increases (decreases) several times, the second value increases (decreases) by the same amount.
Inversely proportional relationship– a relationship in which as one value increases (decreases) several times, the second value decreases (increases) by the same amount.

III. Setting a learning task

– What problem is facing us? (Learn to distinguish between direct and inverse dependencies)
- This - target our lesson. Now formulate topic lesson. (Direct and inverse proportional relationship).
- Well done! Write down the topic of the lesson in your notebooks. (The teacher writes the topic on the board.)

IV. "Discovery" of new knowledge(10 min)

Let's look at problems No. 199.

1. The printer prints 27 pages in 4.5 minutes. How long will it take it to print 300 pages?

27 pages – 4.5 min.
300 pages - x?

2. The box contains 48 packs of tea, 250 g each. How many 150g packs of this tea will you get?

48 packs – 250 g.
X? – 150 g.

3. The car drove 310 km, using 25 liters of gasoline. How far can a car travel on a full 40L tank?

310 km – 25 l
X? – 40 l

4. One of the clutch gears has 32 teeth, and the other has 40. How many revolutions will the second gear make while the first one makes 215 revolutions?

32 teeth – 315 rev.
40 teeth – x?

To compile a proportion, one direction of the arrows is necessary; for this, in inverse proportionality, one ratio is replaced by the inverse.

At the board, students find the meaning of quantities; on the spot, students solve one problem of their choice.

– Formulate a rule for solving problems with direct and inverse proportional dependence.

A table appears on the board:

V. Primary consolidation in external speech(10 min)

Worksheet assignments:

From 21 kg of cottonseed, 5.1 kg of oil was obtained.
How much oil will be obtained from 7 kg of cottonseed?

To build the stadium, 5 bulldozers cleared the site in 210 minutes. How long would it take 7 bulldozers to clear this site?VI. Independent work with self-test according to the standard

(5 minutes)
Two students complete task No. 225 independently on hidden boards, and the rest - in notebooks. They then check the algorithm's work and compare it with the solution on the board. Errors are corrected and their causes are determined. If the task is completed correctly, then the students put a “+” sign next to them.

Students who make mistakes in independent work can use consultants.№ 271, № 270.

VII. Inclusion in the knowledge system and repetition

Six people work at the board. After 3-4 minutes, students working at the board present their solutions, and the rest check the assignments and participate in their discussion.

VIII. Reflection on activity (lesson summary)
– What new did you learn in the lesson?
-What did they repeat?
– What is the algorithm for solving proportion problems?
– Have we achieved our goal?

– How do you evaluate your work?

Example

1.6 / 2 = 0.8;

4 / 5 = 0.8; 5.6 / 7 = 0.8, etc. Proportionality factor

A constant relationship of proportional quantities is called

A constant relationship of proportional quantities is called proportionality factor . The proportionality coefficient shows how many units of one quantity are per unit of another. Direct proportionality

- functional dependence, in which a certain quantity depends on another quantity in such a way that their ratio remains constant. In other words, these variables change

proportionally(x) = , in equal shares, that is, if the argument changes twice in any direction, then the function also changes twice in the same direction.x,, in equal shares, that is, if the argument changes twice in any direction, then the function also changes twice in the same direction. = Mathematically, direct proportionality is written as a formula:faco

Inverse proportionality

n s

Inverse proportionality

- this is a functional dependence, in which an increase in the independent value (argument) causes a proportional decrease in the dependent value (function).

Mathematically, inverse proportionality is written as a formula:

Function properties:
Sources

See what “Direct proportionality” is in other dictionaries:

direct proportionality- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Energy topics in general EN direct ratio ... Technical Translator's Guide

direct proportionality- tiesioginis proporcingumas statusas T sritis fizika atitikmenys: engl. direct proportionality vok. direkte Proportionalität, f rus. direct proportionality, f pranc. proportionnalité directe, f … Fizikos terminų žodynas

PROPORTIONALITY- (from Latin proportionalis proportionate, proportional). Proportionality. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. PROPORTIONALITY lat. proportionalis, proportional. Proportionality. Explanation 25000... ... Dictionary of foreign words of the Russian language

PROPORTIONALITY- PROPORTIONALITY, proportionality, plural. no, female (book). 1. abstract noun to proportional. Proportionality of parts. Body proportionality. 2. Such a relationship between quantities when they are proportional (see proportional ... Ushakov's Explanatory Dictionary

Proportionality- Two mutually dependent quantities are called proportional if the ratio of their values remains unchanged. Contents 1 Example 2 Proportionality coefficient ... Wikipedia

PROPORTIONALITY- PROPORTIONALITY, and, female. 1. see proportional. 2. In mathematics: such a relationship between quantities in which an increase in one of them entails a change in the other by the same amount. Straight line (with a cut with an increase in one value... ... Ozhegov's Explanatory Dictionary

proportionality- And; and. 1. to Proportional (1 value); proportionality. P. parts. P. physique. P. representation in parliament. 2. Math. Dependence between proportionally changing quantities. Proportionality factor. Direct line (in which with... ... encyclopedic Dictionary

ADMINISTRATION OF THE MUNICIPAL FORMATION "CITY OF SARATOV"

MUNICIPAL EDUCATIONAL INSTITUTION

"SECONDARY EDUCATIONAL SCHOOL No. 95 WITH IN-DEPTH

STUDYING INDIVIDUAL SUBJECTS"

Methodological 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

for a lesson on the topic:

“Direct proportionality and its graph.”

Mathematics teacher Elena Viktorovna Goryunova.

We present to your attention a lesson in 7th grade. The teacher works according to a program compiled on the basis of Model programs of basic general education and the author’s program for general education institutions Yu.N. Makarychev. Algebra.7-9 grades //Collection of programs for algebra grades 7-9. M. Prosveshchenie, 2009 compiled by T.A. Burmistrova. The program corresponds to the algebra textbook by Yu.N. Makarychev, N.G. Mindyuk, K.I. Neshkov., S.B. Suvorova., edited by S.A. Telyakovsky “Algebra 7th grade” (Prosveshcheniye publishing house, 2009).

14 hours are allotted for studying the topic “Functions”, of which 6 hours 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:

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

Educational:

To instill a sense of respect for classmates, attention to words, to promote independence, responsibility, and accuracy when constructing drawings

Achieving these goals is accomplished through a series 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 revised specifically for this lesson, taking into account the topic of the lesson, textbook.

Lesson type and structure:

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

Implementation of training principles:

The following principles were implemented in the lesson:

Science of learning.

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

Consciousness, activity and independence of students were achieved in the form of stimulation of cognitive activity with the help of effective techniques and visual aids (such as showing slides, providing historical facts and information from the life of the mathematician and philosopher R. Descartes, individual printed sheets for students.

The principle of comfort was implemented in the lesson.

Forms and methods of teaching:

During the lesson, various forms of training were used - individual and frontal work, mutual testing. Such forms are more rational for this type of lesson, as they allow the child to develop independent thinking, criticality of thought, the ability to defend his 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 problems.

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

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 objectives set for the lesson were realized, the children applied their knowledge in a new situation, everyone could express their point of view. Using visual aids in the form of presentations and individual printed sheets for students allows you to motivate students at every stage of the lesson and avoid overloading and overtiring students.

Lesson topic:

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

Goals:

Educational:

1. Organize students’ activities on understanding the topic “Direct proportionality and its graph” and primary consolidation: defining direct proportionality and constructing its graph, developing skills in competently constructing graphs

2. Create conditions for creating a system of basic knowledge and skills in students’ memory, stimulate search activity

Educational:

1. Develop analytical-synthesizing thinking (promote the development of observation, the ability to analyze, the development of the ability to classify facts, draw generalizing conclusions).

2. Develop abstract thinking (developing the ability to identify general and essential features, distinguish unimportant features and be distracted from them).

3. Encourage students to self-control and mutual control

Educational:

To instill a sense of respect for classmates, attention to words, to promote independence, responsibility, and accuracy when constructing drawings.

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

Lesson plan:

1. Organizational moment.

2.Lesson motivation.

3. Updating knowledge.

4.Learning new material.

5. Fixing the material.

6. Lesson summary.

During the classes.

1. Organizational moment.

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

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

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

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

Student message:(Slide 2)

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 jurisprudence. At the age of 22, he left France and served as a volunteer officer in the forces of various military leaders who participated in the 13-year war. 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 refuge for quiet work on philosophy and mathematics, in which he was interested since childhood, Descartes settled in Holland in 1629, where he lived almost until the end of his life. All of Descartes' major works on philosophy, mathematics, physics, cosmology and physiology were written by him in Holland.

Descartes' mathematical works are collected in his book "Geometry" (1637). In "Geometry" Descartes gave the foundations of analytical 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 degree, and the sign to denote an infinitely large quantity. For variables and unknown quantities, Descartes adopted the notations x, y, z, and for known and constant quantities -a .b .c, as is known, these notations are used in mathematics to this day. Despite the fact that Descartes did not advance very far in the field of analytical geometry, his works had a decisive influence on the further development of mathematics. For 150 years, mathematics developed along the paths outlined by Descartes.

Let's follow the scientist's advice. 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 propose these words (Slide 3) of R. Descartes as the motto of our lesson: “Respect for others gives a reason to respect oneself.”

2.Motivation.

Let's check in what mood you came to class. Draw a smiley face in the margins.

Take the 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 No. 1 with mathematical terms that we will use in class. Correct any mistakes made in the spelling of these terms. (Slide 4)

Exchange leaves and check if all errors are corrected. (Slide 5) -What did you notice? Which word has no mistakes? (function, schedule)

3. Updating knowledge.

a) We became familiar with the concept of “function” in previous lessons. Let's remember the basic concepts and definitions on this topic.

We also worked with function graphs. Which of the dictation words did we use when working on the topic “Graphs of Functions”? What do they mean?

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

Who can tell us what we will talk about in this lesson? What goals will we set for the lesson? (Slide7)

Write down the number on student sheets and write the topic of the lesson: “Direct proportionality and its graph”

Let's remember the material from previous lessons

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

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

Which formula is different from the others? (Slide)

c) How can you write the formulas 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 of the ways to define a function. Direct proportional dependence is a function.

4.Learning new material.

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

Let's read the rule in the textbook on page 65

What is the scope of this function? (The set of all numbers)

Fixing the material.

Complete the task in sheets No. 4 (Slide) Distribute the formulas into 2 groups in accordance with 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,

Group 1:______________________________________________________________

Group 2:______________________________________________________________

Underline the coefficient of direct proportionality.

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

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

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 constructed a graph of a function in task No. 2, can we call this function pr. proportionality? This means that we have already built a graph of proportionality. The rule is in the textbook on page 67.

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

Fixing the material.

Let's build graph No. 7 on student sheets (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, then how many points are needed to construct it? There is already one (0;0)

We carry out No. 300

Lesson summary. Let's summarize the work in today's lesson (Slide). Everything was done. What have you planned?

Reflection. (Slide)

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