Direct and inverse proportionality 6. Tasks on the topic of direct and inverse proportionality

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 more to the smaller one.

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 term of the proportion, we divide the product of the middle terms by the known extreme term of the proportion:

So, 12 meters cost 1344 rubles.

Answer: 1344 rubles.

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Slides captions:

"Direct and reverse proportional dependencies"Grade 6 Mathematics teacher MAOU "Kurovskaya secondary school No. 6" Chugreeva T. D.

Mathematics is the basis and queen of all sciences, And I advise you to make friends with it, my friend. Her wise laws if you do it, you will increase your knowledge, you will apply it. Can you swim in the sea, Can you fly in space. You can build a house for people: It will stand for a hundred years. Do not be lazy, work, try, Knowing the salt of sciences Try to prove everything, But tirelessly.

Finish the phrase: 1. A direct proportional relationship is such a dependence of quantities at which ... 2. An inverse proportional relationship is such a dependence of quantities at which ... 3. To find the unknown extreme member of the proportion ... 4. The middle member of the proportion is ... 5. The proportion is correct, if ... C) ... when one value increases several times, the other decreases by the same amount. X) ... the product of the extreme terms is equal to the product of the middle terms of the proportion. A) ... when one value is increased several times, the other increases by the same amount. P) ... you need to divide the product of the middle terms of the proportion by the known extreme term. Y) ... when one value is increased several times, the other increases by the same amount. E) ... the ratio of the product of the extreme terms to the known mean.

The growth of the child and his age are directly proportional. 2. With a constant width of a rectangle, its length and area are directly proportional. 3. If the area of ​​the rectangle constant, then its length and width are inversely proportional. 4. The speed of the car and the time of its movement are inversely proportional.

5. The speed of the car and its distance traveled are inversely proportional. 6. The income of the cinema box office is directly proportional to the number of tickets sold, sold at the same price. 7. Carrying capacity of machines and their number are inversely proportional. 8. The perimeter of a square and the length of its side are directly proportional. 9. At a constant price, the cost of a commodity and its mass are inversely proportional.

Come on, pencils aside! No papers, no pens, no chalk! Verbal counting! We do this business Only by the power of the mind and soul! VERBAL COUNTING

Find the unknown term of the proportion? ? ? ? ? ? ?

"DIRECT PROPORTIONAL DEPENDENCE" LESSON TOPIC AND INVERSE

a) A cyclist travels 75 km in 3 hours. How long will it take a cyclist to travel 125 km at the same speed? b) 8 identical pipes fill the pool in 25 minutes. How many minutes will it take 10 such pipes to fill the pool? c) A team of 8 workers completes the task in 15 days. How many workers can complete this task in 10 days, working at the same productivity? d) From 5.6 kg of tomatoes, 2 liters of tomato sauce are obtained. How many liters of sauce can be obtained from 54 kg of tomatoes? Make proportions for solving problems:

Answers: a) 3:x=75:125 b) 8:10= X:2 5 c) 8: x=10: 15 d) 5.6:54=2: X

To heat the school building, coal was harvested for 180 days at a consumption rate of 0.6 tons of coal per day. How many days will this reserve last if it is spent daily at 0.5 tons? Solve the problem

Short record: Mass (t) for 1 day Number of days At the rate of 0.6 180 0.5 x Let's make a proportion: ; ; Answer: 216 days. Decision.

AT iron ore 7 parts of iron account for 3 parts of impurities. How many tons of impurities are in an ore that contains 73.5 tons of iron? #793 Solve the problem

Number of parts Mass Iron 7 73.5 Impurities 3 x; Answer: 31.5 kg of impurities. Decision. ; №793

An unknown number is denoted by the letter x. The condition is written in the form of a table. The type of dependence between quantities is established. Directly proportional dependence is indicated by equally directed arrows, and inversely proportional dependence is indicated by oppositely directed arrows. The proportion is recorded. An unknown member is located. Algorithm for solving problems for direct and inverse proportionality:

Solve the equation:

No. 1. On the way from one village to another at a speed of 12.5 km / h, the cyclist spent 0.7 hours. At what speed did he have to go to cover this path in 0.5 hours? No. 2. From 5 kg of fresh plums, 1.5 kg of prunes are obtained. How many prunes will be obtained from 17.5 kg of fresh plums? No. 3. The car drove 500 km, having spent 35 liters of gasoline. How many liters of gasoline do you need to travel 420 km? No. 4. 12 crucians were caught in 2 hours. How many carp will be caught in 3 hours? #5 Six painters can do some work in 18 days. How many more painters need to be invited to complete the job in 12 days? Independent work Solve problems by making proportions.

Solving problems from independent work Solution: No. 1 Short entry: Speed ​​(km / h) Time (h) 12.5 0.7 x 0.5 Answer: 17.5 km / h Solution: No. 2 Short entry: Plums (kg ) Prunes (kg) 5 1.5 17.5 x; ; kg Answer: 5.25 kg; ; ;

Solving problems from independent work Solution: No. 3 Solution: No. 5 Brief record: Brief record: Distance (km) Gasoline (l) 500 35 420 x; Answer: 29.4 liters. Number of babies Time (days) 6 18 x 12; ; painters will complete the work in 12 days. 1) 9 -6 = 3 painters still need to be invited. Answer: 3 painters.

Additional task: #6. A mining enterprise needs to purchase 5 new machines for a certain amount of money at a price of 12 thousand rubles. for one. How many such cars can the enterprise buy if the price for one car becomes 15 thousand rubles? Decision: No. 1 Brief entry: Number of cars (pcs) Price (thousand rubles) 5 12 x 15; cars. ; Answer: 4 cars.

Home rear No. 812 No. 816 No. 818

Thank you for the lesson!

Preview:

Chugreeva Tatyana Dmitrievna 206818644

Math lesson in 6th grade

on the topic "Direct and inverse proportional relationships"

Developed
mathematic teacher
MAOU "Kurovskaya secondary school No. 6"
Chugreeva Tatyana Dmitrievna

Lesson Objectives:

educational- update the concept of "dependence" between quantities;

Educational through problem solving, setting additional questions and tasks to develop creative and mental activity students;

Independence;

self-esteem skills;

Educational- to cultivate interest in mathematics as a part of human culture.

Equipment: TCO necessary for the presentation: a computer and a projector, sheets for recording answers, cards for the reflection stage (three each), a pointer.

Lesson type: a lesson in the application of knowledge.

Lesson organization forms:frontal, collective, individual work.

During the classes

1. Organizing time.
   

The teacher reads: (slide number 2)

Mathematics is the basis and queen of all sciences,
And I advise you to make friends with her, my friend.
Her wise laws, if you follow,
Increase your knowledge
You will be using them.
Can you swim in the sea
You can fly in space.
You can build a house for people:
It will stand for a hundred years.
Don't be lazy, work hard
Knowing the salt of sciences.
Try to prove everything
But don't give up.

2. Checking the studied material.

1. Finish the sentence:(slide 3). (Children first complete the task on their own, writing down only the letters corresponding to the correct answer on the sheets. Then they raise their hand. After that, the teacher reads the question aloud, and the students answer).

1. A direct proportional relationship is such a dependence of quantities in which ...
2. An inverse proportional relationship is such a dependence of quantities at which ...
3. To find the unknown extreme term of the proportion...
4. The middle term of the proportion is...
5. The proportion is correct if...

C) ... when one value increases several times, the other decreases by the same amount.

X) ... the product of the extreme terms is equal to the product of the middle terms of the proportion.

A) ... when one value is increased several times, the other increases by the same amount.

P) ... you need to divide the product of the middle terms of the proportion by the known extreme term.

Y) ... when one value is increased several times, the other increases by the same amount.

E) ... the ratio of the product of the extreme terms to the known mean.

Answer: SUCCESS. (slide 6)

1. Oral counting: (slides 6-7)

Come on, pencils aside!

No papers, no pens, no chalk!

Verbal counting! We're doing this thing

Only by the power of the mind and soul!

Exercise: Find the unknown term of the proportion:

Answers: 1) 39; 24; 3; 24; 21.

2)10; 3; 13.

1. The topic of the lesson. slide number 8 (Provides motivation for students to learn.)

• The topic of our lesson is “Direct and inverse proportional relationships”.
• In previous lessons, we considered direct and inverse proportional dependence of quantities. Today in the lesson we will decide different tasks using a proportion, establishing the type of relationship between the data. Let's repeat the main property of proportions. And the next lesson, concluding on this topic, i.e. lesson - control work.

1. The stage of generalization and systematization of knowledge.

1) Task1.

Make proportions for solving problems:(work in notebooks)

a) A cyclist travels 75 km in 3 hours. How long will it take a cyclist to travel 125 km at the same speed?

b) 8 identical pipes fill the pool in 25 minutes. How many minutes will it take 10 such pipes to fill the pool?

c) A team of 8 workers completes the task in 15 days. How many workers can complete this task in 10 days, working at the same productivity?

d) From 5.6 kg of tomatoes, 2 liters of tomato sauce are obtained. How many liters of sauce can be obtained from 54 kg of tomatoes?

Check answers. (Slide number 10) (self-assessment: put + or - in pencil innotebooks; analyze errors)

Answers: a) 3:x=75:125 c) 8:x=10:15

b) 8:10= X:2 5 d) 5.6:54=2: X

Solve the problem

№788 (p. 130, Vilenkin's textbook)(after parsing by yourself)

In the spring, during the greening of the city, lindens were planted on the street. 95% of the milestones of planted lindens were accepted. How many lindens were planted if 57 lindens were taken?

• Read the task.
• What two quantities are mentioned in the problem?(about the number of limes and their percentages)
• What is the relationship between these quantities?(directly proportional)
• Compose short note, proportion and solve the problem.

Decision:

	Lindens (pcs.)	Percentage %
planted
Accepted

; ; x=60.

Answer: 60 lindens were planted.

Solve the problem: (slide No. 11-12) (after parsing, decide on your own; mutual check, then the solution is displayed on the screen slide No. 23)

To heat the school building, coal was harvested for 180 days at a consumption rate of 0.6 tons of coal per day. How many days will this reserve last if it is spent daily at 0.5 tons?

Decision:

Brief entry:

	Weight (t) for 1 day	Quantity days
According to the norm

Let's make a proportion:

; ; days

Answer: 216 days.

No. 793 (p. 131) (field parsing by yourself; self-control.

(Slide number 13)

In iron ore, 7 parts of iron account for 3 parts of impurities. How many tons of impurities are in an ore that contains 73.5 tons of iron?

Solution: (slide number 14)

	Quantity parts	Weight
Iron		73,5
impurities

Answer: 31.5 kg of impurities.

So, let's formulate an algorithm for solving problems using proportions.

Algorithm for solving problems directly

and inversely proportional relationships:

1. An unknown number is denoted by the letter x.
2. The condition is written in the form of a table.
3. The type of dependence between quantities is established.
4. Directly proportional dependence is indicated by equally directed arrows, and inversely proportional dependence is indicated by oppositely directed arrows.
5. The proportion is recorded.
6. An unknown member is located.

Repetition of the studied material.

No. 763 (i) (p. 125) (with commentary at the board)

6. Stage of control and self-control of knowledge and methods of action.
(slide №17-19)

Independent work(10 - 15 min) (Mutual check: on the finished slides, students check each other independent work, while setting + or -. The teacher at the end of the lesson collects notebooks for review).

Solve problems by making proportions.

No. 1. On the way from one village to another at a speed of 12.5 km / h, the cyclist spent 0.7 hours. At what speed did he have to go to cover this path in 0.5 hours?

Decision:

Brief entry:

	Speed ​​(km/h)	Time (h)
	12,5

Let's make a proportion:

; ; km/h

Answer: 17.5 km/h

No. 2. From 5 kg of fresh plums, 1.5 kg of prunes are obtained. How many prunes will be obtained from 17.5 kg of fresh plums?

Decision:

Brief entry:

	Plums (kg)	Prunes (kg)
	17,5

Let's make a proportion:

; ; kg

Answer: 5.25 kg

No. 3. The car drove 500 km, having spent 35 liters of gasoline. How many liters of gasoline do you need to travel 420 km?

Decision:

Brief entry:

	Distance (km)	Gasoline (l)

Let's make a proportion:

; ; l

Answer: 29.4 liters.

№4 . 12 crucians were caught in 2 hours. How many carp will be caught in 3 hours?

Answer: the answer does not exist. these quantities are neither directly proportional nor inversely proportional.

№5 Six painters can do some work in 18 days. How many more painters need to be invited to complete the job in 12 days?

Decision:

Brief entry:

	Number of painters	Time (days)

Let's make a proportion:

; ; painters will complete the work in 12 days.

1) 9 -6=3 painters still need to be invited.

Answer: 3 painters.

Additional (slide number 33)

No. 6. A mining enterprise needs to purchase 5 new machines for a certain amount of money at a price of 12 thousand rubles. for one. How many such cars can the enterprise buy if the price for one car becomes 15 thousand rubles?

Decision:

Brief entry:

	Number of machines (pcs.)	Price (thousand rubles)

Let's make a proportion:

; ; cars.

Answer: 4 cars.

1. The stage of summing up the lesson

• What did we learn in the lesson?(The concepts of direct and inverse proportional dependence of two quantities)
• Give examples of directly proportional quantities.
• Give examples of inversely proportional quantities.
• Give examples of quantities whose dependence is neither directly nor inversely proportional.

1. Homework (slide 21)
   № 812, 816, 818.

Thanks for the lesson slide number 22

Proportionality is the relationship between two quantities, in which a change in one of them entails a change in the other by the same amount.

Proportionality is direct and inverse. AT this lesson we will look at each of them.

Lesson content

Direct proportionality

Suppose a car is moving at a speed of 50 km/h. We remember that speed is the distance traveled per unit of time (1 hour, 1 minute or 1 second). In our example, the car is moving at a speed of 50 km / h, that is, in one hour it will travel a distance equal to fifty kilometers.

Let's plot the distance traveled by the car in 1 hour.

Let the car drive for another hour at the same speed of fifty kilometers per hour. Then it turns out that the car will travel 100 km

As can be seen from the example, doubling the time led to an increase in the distance traveled by the same amount, that is, twice.

Quantities such as time and distance are said to be directly proportional. The relationship between these quantities is called direct proportionality.

Direct proportionality is the relationship between two quantities, in which an increase in one of them entails an increase in the other by the same amount.

and vice versa, if one value decreases by a certain number of times, then the other decreases by the same amount.

Let's assume that it was originally planned to drive a car 100 km in 2 hours, but after driving 50 km, the driver decided to take a break. Then it turns out that by reducing the distance by half, the time will decrease by the same amount. In other words, a decrease in the distance traveled will lead to a decrease in time by the same factor.

An interesting feature of directly proportional quantities is that their ratio is always constant. That is, when changing the values ​​of directly proportional quantities, their ratio remains unchanged.

In the considered example, the distance was at first equal to 50 km, and the time was one hour. The ratio of distance to time is the number 50.

But we have increased the time of movement by 2 times, making it equal to two hours. As a result, the distance traveled increased by the same amount, that is, it became equal to 100 km. The ratio of one hundred kilometers to two hours is again the number 50

The number 50 is called direct proportionality coefficient. It shows how much distance there is per hour of movement. AT this case the coefficient plays the role of the speed of movement, since the speed is the ratio of the distance traveled to the time.

Proportions can be made from directly proportional quantities. For example, the ratios and make up the proportion:

Fifty kilometers are related to one hour as one hundred kilometers are related to two hours.

Example 2. The cost and quantity of the purchased goods are directly proportional. If 1 kg of sweets costs 30 rubles, then 2 kg of the same sweets will cost 60 rubles, 3 kg - 90 rubles. With the increase in the cost of the purchased goods, its quantity increases by the same amount.

Since the value of a commodity and its quantity are directly proportional, their ratio is always constant.

Let's write down the ratio of thirty rubles to one kilogram

Now let's write down what the ratio of sixty rubles to two kilograms is equal to. This ratio will again be equal to thirty:

Here, the direct proportionality coefficient is the number 30. This coefficient shows how many rubles per kilogram of sweets. AT this example the coefficient plays the role of the price of one kilogram of goods, since the price is the ratio of the cost of the goods to its quantity.

Inverse proportionality

Consider the following example. The distance between the two cities is 80 km. The motorcyclist left the first city, and at a speed of 20 km/h reached the second city in 4 hours.

If the speed of a motorcyclist was 20 km/h, this means that every hour he traveled a distance equal to twenty kilometers. Let us depict in the figure the distance traveled by the motorcyclist and the time of his movement:

On the way back the speed of the motorcyclist was 40 km/h, and he spent 2 hours on the same journey.

It is easy to see that when the speed changes, the time of movement has changed by the same amount. And it changed in reverse side- that is, the speed increased, and the time, on the contrary, decreased.

Quantities such as speed and time are called inversely proportional. The relationship between these quantities is called inverse proportionality.

Inverse proportionality is the relationship between two quantities, in which an increase in one of them entails a decrease in the other by the same amount.

and vice versa, if one value decreases by a certain number of times, then the other increases by the same amount.

For example, if on the way back the speed of a motorcyclist was 10 km / h, then he would cover the same 80 km in 8 hours:

As can be seen from the example, a decrease in speed led to an increase in travel time by the same factor.

The peculiarity of inversely proportional quantities is that their product is always constant. That is, when changing the values ​​of inversely proportional quantities, their product remains unchanged.

In the considered example, the distance between the cities was 80 km. When changing the speed and time of the motorcyclist, this distance always remained unchanged.

A motorcyclist could cover this distance at a speed of 20 km/h in 4 hours, and at a speed of 40 km/h in 2 hours, and at a speed of 10 km/h in 8 hours. In all cases, the product of speed and time was equal to 80 km

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Slides captions:

Definition, examples, tasks Direct and inverse proportionality S v t Price Quantity Cost Number of workers Productivity Amount of work

Example 2 Example 1 The concept of direct and inverse proportionality Misha walked with constant speed 4 km/h How far will he travel in 1; 3; 6; 10 hours? Time and distance are proportional values. The more hours Misha goes, the more distance he will cover. t 1 3 6 10 S Misha traveled a distance of 36 km. With what speed did he move if he arrived for 1; 2; 3; 6 hours? Time and distance are proportional values. The more hours Misha goes, the slower the speed of movement. t 1 2 3 6 V Are the values ​​in examples 1 and 2 proportional? Is the same proportionality shown in the examples?

Definition 2 Definition 1 Definition of direct and inverse proportionality Two quantities are called directly proportional if, when one of them increases (decreases) several times, the other also increases (decreases) by the same amount. Vel. 1 - Lead 2 Lead 1. - Lead 2. Lead. 1 - Lead 2 Lead 1. - Lead 2. Two quantities are called directly proportional if, with an increase (decrease) in one of them several times, the other decreases (increases) by the same amount. Vel. 1 - Lead 2 Lead 1. - Lead 2.

Determination of direct and inverse proportionality For 5 notebooks in a cage, they paid 40 rubles. How much will they pay for 12 of the same notebooks? It took 18 m of fabric to sew 9 shirts. How many shirts will you get from 14 meters? Determine the type of proportionality 6 workers will complete the work in 5 hours, how long will it take 3 workers to do this work? The tailor has a piece of cloth. If he sews dresses from it, each of which takes 2 meters, then 15 dresses will be obtained. How many suits can come out of the same cut if each suit takes 3 meters of fabric?

Definition of direct and inverse proportionality Make a short note and determine the type of proportionality. (The values ​​of the same name are written one below the other) Make a proportion. If direct proportionality, then the values ​​\u200b\u200bare written in proportion without change. If it is inversely proportional, then in one of the values ​​the data are interchanged (vice versa). The unknown term of the proportion is found. Algorithm for solving the problem For 5 notebooks in a cage, they paid 40 rubles. How much will they pay for 12 of the same notebooks? Quantity Cost of 5 notebooks - 40 rubles. 12 notebooks - x rub. Answer: 96 rubles.

Definition of direct and inverse proportionality Make a short note and determine the type of proportionality. (The values ​​of the same name are written one below the other) Make a proportion. If direct proportionality, then the values ​​\u200b\u200bare written in proportion without change. If it is inversely proportional, then in one of the values ​​the data are interchanged (vice versa). The unknown term of the proportion is found. Algorithm for solving the problem 6 workers will complete the work in 5 hours, how long will it take 3 workers to do this work? Quantity Time 6 working – 5 hours. 3 working hours. Answer: 10 hours.

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