6 proportional. Direct and inverse proportionality

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Slide captions:

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

Mathematics is the basis and queen of all sciences, And I advise you to make friends with her, my friend. If you follow her wise laws, you will increase your knowledge, you will begin to apply them. You can swim on the sea, You can 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 the sciences Try to prove everything, But tirelessly.

Finish the phrase: 1. Direct proportional dependence is such a dependence of quantities at which ... 2. Inverse proportional dependence 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) ... with an increase in one value by 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) ... with an increase in one value 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 extreme members to the known mean.

A child's height and age are directly proportional. 2. With a constant width of a rectangle, its length and area are directly proportional. 3. If the area of \u200b\u200bthe rectangle is constant, then its length and width are inversely proportional. 4. Vehicle speed and travel time are inversely proportional.

5. Vehicle speed and distance traveled are inversely proportional. 6. The revenue 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 value of the goods and its weight are inversely proportional.

Well, pencils aside! No papers, no pens, no chalk!

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Solving problems from independent work Solution: No. 1 Brief note: Speed \u200b\u200b(km / h) Time (h) 12.5 0.7 x 0.5 Answer: 17.5 km / h Solution: No. 2 Brief note: 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 note: Brief note: Distance (km) Gasoline (l) 500 35 420 x; Answer: 29.4 liters. Number of malays Time (days) 6 18 x 12; ; painters will do the job in 12 days. 1) 9 -6 \u003d 3 painters still need to be invited. Answer: 3 painters.

Additional task: # 6. The 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 of these cars can a company buy if the price for one car becomes 15 thousand rubles? Solution: 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 Tatiana Dmitrievna 206818644

6th grade math lesson

on the topic "Direct and inverse proportional dependencies"

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

Lesson objectives:

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

Developing - through the solution of problems, the formulation of additional questions and tasks, to develop the creative and mental activity of students;

Independence;

Self-assessment skills;

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

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

Lesson type: lesson in the application of knowledge.

Lesson organization forms: frontal, collective, individual work.

During the classes

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
You will increase your knowledge,
You will apply them.
You can swim on 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, try
Learning the salt of the sciences.
Try to prove everything
But tirelessly.

2. Verification of the studied material.

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

Direct proportional dependence is such a dependence of quantities at which ...
An inverse proportional relationship is such a dependence of quantities at which ...
To find the unknown extreme term of the proportion ...
The middle term of the proportion is equal to ...
The proportion is correct if ...

C)… with an increase in one value by 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) ... with an increase in one value 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 extreme members to the known mean.

Answer: SUCCESS. (slide 6)

Verbal counting: (slides 6-7)

Well, pencils aside!

No papers, no pens, no chalk!

Verbal counting! We do this business

Only by the power of the mind and soul!

The task: Find the unknown member of the proportion:

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

2)10; 3; 13.

Lesson topic message.slide number 8 (Provides student learning motivation.)

The topic of our lesson is "Direct and inverse proportional dependencies."
In the previous lessons, we considered direct and inverse proportional dependence of values. Today in the lesson we will solve different problems using proportions, establishing the type of connection between data. Let's repeat the main property of proportions. And the next lesson, completing on this topic, i.e. lesson - test.

The stage of generalization and systematization of knowledge.

1) Task 1.

Compose proportions for solving problems:(work in notebooks)

a) A cyclist travels 75 km in 3 hours. How long will it take for 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 will be able to 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 you get from 54 kg of tomatoes?

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

Answers: a) 3: x \u003d 75: 125 c) 8: x \u003d 10: 15

b) 8: 10 \u003d X: 2 5 d) 5.6: 54 \u003d 2: X

Solve the problem

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

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

Read the problem.
What two quantities are mentioned in the problem?(about the number of limes and their percentage)
What is the relationship between these values?(directly proportional)
Make a short note, proportion and solve the problem.

Decision:

	Lindens (pcs.)	Interest%
Planted
Accepted

; ; x \u003d 60.

Answer: 60 lindens were planted.

Solve the problem: (slide number 11-12) (after parsing, solve by yourself; mutual check, then the solution is displayed on the screen slide number 23)

Coal was prepared for heating the school building for 180 days at a consumption rate of 0.6 tons of coal per day. How many days will this stock last if you spend 0.5 tonnes daily?

Decision:

Brief entry:

Weight (t)

in 1 day

amount

days

At the norm

Let's make the proportion:

; ; days

Answer: 216 days.

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

(Slide number 13)

In iron ore, 7 parts of iron have 3 parts of impurities. How many tons of impurities are in the ore, which contains 73.5 tons of iron?

Solution: (slide number 14)

amount

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 in a straight line

and inverse proportional relationships:

The unknown number is indicated by the letter x.
The condition is written in the form of a table.
The type of relationship between values \u200b\u200bis established.
Directly proportional dependence is indicated by equally directed arrows, and inversely proportional dependence is indicated by oppositely directed arrows.
The proportion is recorded.
Its unknown member is located.

Repetition of the material studied.

No. 763 (i) (p. 125) (with comments at the blackboard)

6. The stage of control and self-control of knowledge and methods of action.
(slide number 17-19)

Independent work(10 - 15 min.) (Mutual check: according to the finished slides, students check each other independent workwhile exposing + or -. The teacher at the end of the lesson collects notebooks for viewing).

Solve problems by proportioning.

# 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 travel to overcome this path in 0.5 hours?

Decision:

Brief entry:

	Speed \u200b\u200b(km / h)	Time (h)
	12,5

Let's make the proportion:

; ; km / h

Answer: 17.5 km / h

# 2. From 5 kg of fresh plums, 1.5 kg of prunes are obtained. How many prunes do you get from 17.5 kg of fresh plums?

Decision:

Brief entry:

	Plums (kg)	Prunes (kg)

	17,5

Let's make the proportion:

; ; kg

Answer: 5.25 kg

No. 3. The car drove 500 km using 35 liters of gasoline. How many liters of gasoline will it take to travel 420 km?

Decision:

Brief entry:

	Distance (km)	Gasoline (l)

Let's make the proportion:

; ; l

Answer: 29.4 liters.

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

Answer: there is no answer because 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 get the job done in 12 days?

Decision:

Brief entry:

	Number of painters	Time (days)

Let's make the proportion:

; ; painters will do the job in 12 days.

1) 9 -6 \u003d 3 painters still need to be invited.

Answer: 3 painters.

Additional (slide number 33)

No. 6. The 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 a company buy if the price for one car becomes 15 thousand rubles?

Decision:

Brief entry:

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

Let's make the proportion:

; ; cars.

Answer: 4 cars.

The stage of summing up the lesson

What did we learn in the lesson?(The concept of direct and inverse proportional relationship of two quantities)
Give examples of directly proportional quantities.
Give examples of inverse proportions.
Give examples of quantities for which the relationship is neither directly nor inversely proportional.

Homework (slide 21)
№ 812, 816, 818.

Thank you for the lesson slide number 22

The two quantities are called directly proportionalif when one of them is increased several times, the other increases by the same amount. Accordingly, when one of them decreases several times, the other decreases by the same amount.

The relationship between such quantities is a direct proportional relationship. Examples of direct proportional dependence:

1) at a constant speed, the distance traveled is directly proportional to time;

2) the perimeter of the square and its side are directly proportional values;

3) the cost of a product purchased at one price is directly proportional to its quantity.

To distinguish direct proportional dependence from the inverse, you can use the proverb: "The further into the forest, the more firewood."

It is convenient to solve problems with directly proportional quantities using proportion.

1) To make 10 parts, you need 3.5 kg of metal. How much metal will be used to make 12 of these parts?

(We reason like this:

1. In the filled column, put the arrow in the direction from the largest number to the smallest.

2. The more parts, the more metal is needed to make them. This means that this is a directly proportional relationship.

Let x kg of metal be needed to make 12 parts. We make the proportion (in the direction from the beginning of the arrow to its end):

12: 10 \u003d x: 3.5

To find, it is necessary 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) 1,680 rubles were paid for 15 meters of fabric. How much does 12 meters of such fabric cost?

(1. In the filled column, put the arrow in the direction from the largest number to the smallest.

2. The less fabrics are bought, the less you have to pay for them. This means that this is a directly proportional relationship.

3. Therefore, the second arrow is equally directed with the first).

Let x rubles cost 12 meters of fabric. We make the proportion (from the beginning of the arrow to its end):

15: 12 \u003d 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:

This means that 12 meters cost 1,344 rubles.

Answer: 1344 rubles.

If a machine with numerical control in 2 hours produces 28 parts, then in twice as long, that is, in 4 hours, it will produce twice as many such parts, that is, 28 2 \u003d 56 parts. How many times more time the machine will work, so many times more parts it will produce. This means that the ratios 4: 2 and 56: 28 are equal. Therefore, the proportion 4: 2 \u003d 56: 28 is correct. Such quantities as the operating time of the machine and the number of manufactured parts are called directly proportional quantities.

If two quantities are directly proportional, then the ratios of the corresponding values \u200b\u200bof these quantities are equal.

Let the train travel from city A to city B at a speed of 40 km / h in 12 hours.If the speed is doubled, i.e., to make it equal to 80 km / h, then the train will spend half the time on the same route, i.e. That is, 6 hours. How many times the speed of movement increases, the time of movement will decrease by the same amount. In this case, the ratio 80: 40 will be equal not to the ratio 6: 12, but to the inverse ratio of 12: 6. Therefore, the ratio 80: 40 \u003d 12: 6. Such quantities as speed and time are called inversely proportional quantities.

If the quantities are inversely proportional, then the ratio of the values \u200b\u200bof one quantity is equal to the inverse ratio of the corresponding values \u200b\u200bof the other quantity.

Not all two quantities are directly proportional or inversely proportional. For example, a child's height increases with increasing age, but these values \u200b\u200bare not proportional, since when the age is doubled, the child's height does not double.

Proportional value problems can be solved using proportion.

Problem 1. For 3.2 kg of goods paid 115.2 rubles. How much should I pay for 1.5 kg of this item?

Decision. Let us briefly write down the condition of the problem in the form of a table, denoting with the letter x the cost (in rubles) of 1.5 kg of this product.

The entry will look like this:

The relationship between the quantity of goods and the cost of purchase is directly proportional, since if you buy several times more goods, then the cost of the purchase will increase by the same amount. We will conventionally designate such dependence by equally directed arrows.

Let's write down the proportion:.

Answer: 54 p.

Problem 2. Two rectangles have the same area. The first rectangle is 3.6 m long and 2.4 m wide. The second rectangle is 4.8 m long. Find the width of the second rectangle.

Decision. Having denoted the width (in meters) of the second rectangle with the letter x, we briefly write the condition of the problem:

The relationship between the width and length at the same value of the area of \u200b\u200bthe rectangle is inversely proportional, since if the length of the rectangle is increased several times, then the width must be reduced by the same amount. Let's conventionally designate such dependence by oppositely directed arrows.

Let's write the proportion:

Now let's find the unknown member of the proportion:

Answer: 1.8 m.

Self-test questions

What quantities are called directly proportional? What can be said about the ratios of the corresponding values \u200b\u200bof such quantities?
Give examples of directly proportional quantities.
What quantities are called inverse proportional? What can be said about the ratios of the corresponding values \u200b\u200bof such quantities?
Give examples of inverse proportions.
Give examples of quantities for which the relationship is neither directly nor inversely proportional.

Exercise

782. Determine whether the relationship between the quantities is directly proportional, inversely proportional, or not proportional:

a) the way traveled by the car with constant speed, and the time of its movement;
b) the cost of goods purchased at one price and their quantity;
c) the area of \u200b\u200bthe square and the length of its side;
d) the mass of the steel bar and its volume;
e) the number of workers performing some work with the same productivity, and the time spent on this work;
f) the value of the goods and their quantity purchased for a certain amount of money;
g) the age of the person and the size of his shoes;
h) the volume of the cube and the length of its ribs;
i) the perimeter of the square and the length of its side;
j) a fraction and its denominator, if the numerator does not change;
k) a fraction and its numerator, if the denominator does not change.

Solve Problems No. 783 - 794 by proportion.

783. A steel ball with a volume of 6 cm 3 has a mass of 46.8 g. What is the mass of a ball made of the same steel if its volume is 2.5 cm 3?

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

785. For the construction of the stadium, 5 bulldozers cleared the site in 210 minutes. How long would it take 7 bulldozers to clear this area?

786. To transport the cargo, it took 24 vehicles with a lifting capacity of 7.5 tons. How many vehicles with a lifting capacity of 4.5 tons are needed to transport the same cargo?

787. Peas were sown to determine seed germination. Out of 200 sown peas, 170 have sprouted. What percentage of peas sprouted (germination percentage)?

788. In the spring, linden trees were planted on the street during the greening of the city. Accepted 95% of all planted lindens. How many lindens were planted if 57 lindens were taken?

789. There are 80 students in the ski section. Among them are 32 girls. What percentage of the section participants are girls and what percentage are boys?

790. The plant was supposed to melt 980 tons of steel within a month. But the plan was fulfilled by 115%. How many tons of steel did the plant melt?

791. In 8 months, the worker fulfilled 96% of the annual plan. What percentage of the annual plan will be fulfilled by the worker in 12 months if he works with the same productivity?

792. In three days, 16.5% of all beets were harvested. How many days will it take to harvest 60.5% of all beets with the same production rate?

793. In iron ore, 7 parts of iron have 3 parts of impurities. How many tons of impurities are in the ore that contains 73.5 tons of iron?

794. To prepare borscht, for every 100 g of meat, you need to take 60 g of beets. How many beets should you take for 650 g of meat?

795. Calculate orally:

796. Present as the sum of two fractions with numerator 1 each of the following fractions: .

797. From the numbers 3, 7, 9 and 21, make two correct proportions.

798. The middle terms are 6 and 10. What can be the extreme terms? Give examples.

799. At what value of x is the proportion correct:

800. Find the relationship:

a) 2 min to 10 s;
b) 0.3 m 2 to 0.1 dm 2;
c) 0.1 kg to 0.1 g;
d) 4 hours to 1 day;
e) 3 dm 3 to 0.6 m 3.

801. Where should the number c be located on the coordinate ray for the proportion to be correct (fig. 34)?

Figure: 34

802. Develop your memory! Cover the table with a piece of paper. Open the first line for a few seconds and then, closing it again, try to repeat or write down the three numbers of this line. If you have reproduced all the numbers correctly, go to the second line of the table. If a mistake is made in any line, write several sets of the same number of two-digit numbers as in the line and practice memorizing them. If you can reproduce at least five two-digit numbers without error, you have a good memory.

803. Solve the equation:

804. Is it possible to make the correct proportion from the following numbers:

805. From the equality of the products 3 24 \u003d 8 9, make up three correct proportions.

806. The length of the segment AB is 8 dm, and the length of the segment CD is 2 cm. Find the ratio of the lengths of the segments AB and CD. What part of the length of the segment AB is the length of the segment CD?

807. There are 460 vacationers in the sanatorium, of which 70% are adults, and the rest are children. How many children were vacationing in the sanatorium?

808. Find the meaning of the expression:

809. Solve the problem:

When processing a part from a casting weighing 40 kg, 3.2 kg was spent as waste. What percentage is the mass of the part from the mass of the casting?
When sorting grain from 1750 kg, 105 kg were used as waste. What percentage of grain is left?

810. Find the meaning of the expression:

6,0008: 2,6 + 4,23 0,4;
2,91 1,2 + 12,6288: 3,6.

811. From 20 kg of apples, 16 kg of applesauce is obtained. How many applesauce will be made from 45 kg of apples?

812. Three painters can complete the job in 5 days. To speed up the work, two more painters were added. How long will it take for them to finish the job if all painters work with the same performance?

813. A concrete slab with a volume of 2.5 m 3 has a mass of 4.75 tons. What is the volume of a slab of the same concrete if its mass is 6.65 tons?

814. Sugar beets contain 18.5% sugar. How much sugar is contained in 38.5 tons of sugar beets? Round your answer to tenths of a ton.

815. The sunflower seeds of the new variety contain 49.5% oil. How many kilograms of such seeds should be taken to contain 29.7 kg of oil?

816. 80 kg of potatoes contain 14 kg of starch. Find the percentage of starch in such potatoes.

817. Flax seeds contain 47% oil. How much oil is in 80 kg of flax seeds?

818. Rice contains 75% starch and barley 60%. How much barley should you take so that it contains the same amount of starch as 5 kg of rice contains?

819. Find the meaning of the expression:

a) 203.81: (141 - 136.42) + 38.4: 0.75;
b) 96: 7.5 + 288.51: (80 - 76.74).

6th grade math lesson

on the topic "Direct and inverse proportional dependencies"

Developed
mathematic teacher
MOU "Mikhailovskaya Secondary School named after
Hero Soviet Union V.F. Nesterov "
Kleymenova D.M.

Lesson objectives :

1. Didactic :

to contribute to the formation and consolidation of skills and abilities for solving problems using proportions;

to teach to distinguish two quantities in the conditions of tasks and establish the type of relationship between them;

write a short note and make up the proportion;

to consolidate the skills and abilities to solve equations that look like proportions.

2. Developing :

develop memory, attention, continue the development of students' mathematical speech;

promote the development of students' creative activity and interest in the subject of mathematics.

3. Educational :

to educate accuracy, to form an interest in mathematics;

foster the ability to listen carefully to the opinions of others, foster self-confidence, foster a culture of communication.

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

Lesson type: lesson in the application of knowledge.

Lesson organization forms: frontal, collective, individual work.

Lesson structure:

Organizational moment, greetings, wishes.

Checking the studied material.

Lesson topic message.

Repetition of the material studied.

The stage of control and self-control of knowledge and methods of action.

The stage of summing up the results of the lesson.

Homework.

Reflection.

During the classes

Organizing time. (slide 3)
(Greetings, fixing the absent, checking the preparedness of students for educational process, distribution of leaflets and cards for reflection, checking the readiness of the classroom for the lesson, organizing the student's attention).

The teacher reads: (slide number 3)

2. Verification of the studied material.

(identifies problems in the knowledge and methods of activity of students and determines the causes of their occurrence, eliminates the gaps found during the test.)

Oral survey: (slide number 4)

What is called the ratio of two numbers?

How to find a fraction of a number?

What is proportion?

What quantities are called directly proportional?

What does the ratio of two numbers show?

How to find a number by its fraction?

The main property of proportion.

What quantities are called inverse proportional?

Finish the phrase: (slide 5). (Children first complete the task on their own, writing down only letters corresponding to the correct answer on pieces of paper. Then they raise their hand. After that, the teacher reads the question aloud, and the students answer).

Direct proportional dependence is such a dependence of quantities at which ...

An inverse proportional relationship is such a dependence of quantities at which ...

To find the unknown extreme term of the proportion ...

The middle term of the proportion is equal to ...

The proportion is correct if ...

WITH) …when one value is increased 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) ... with an increase in one value 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 extreme members to the known mean.

Answer: SUCCESS.(slide 6)

Graphic dictation (slides 7-10).

Don't say yes and no

And depict with an icon.

"Yes" with "+", no with "-".

(Students work independently. Write down the answers on sheets of paper. Self-test, using slide No. At the end of the lesson, the teacher looks through the sheets)

If the area of \u200b\u200bthe rectangle is constant, then its length and width are inversely proportional.

A child's height and age are directly proportional.

With a constant width of a rectangle, its length and area are directly proportional.

Vehicle speed and travel time are inversely proportional.

Vehicle speed and distance traveled are inversely proportional.

Cinema box office revenue is directly proportional to the number of tickets sold for the same price.

The carrying capacity of machines and their number are inversely proportional.

The perimeter of a square and the length of its side are directly proportional.

At a constant price, the value of the product and its weight are inversely proportional.

Answer: + - + + - + + - -(Slide number 10)

Get an estimate. (Slide number 11)

8 -9 correct answers - "5"

6-7 correct answers - "4"

4-5 correct answers - "3"

Verbal counting: (slides 12-13)

Well, pencils aside!

No papers, no pens, no chalk!

Verbal counting! We do this business

Only by the power of the mind and soul!

The task: Find the unknown member of the proportion:

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

2)10; 3; 13.

Lesson topic message. slide number 14 (Provides student learning motivation.)

The topic of our lesson is "Direct and inverse proportional dependencies."

In the previous lessons, we considered direct and inverse proportional dependence of values. Today in the lesson we will solve different problems using proportions, establishing the type of connection between data. Let's repeat the main property of proportions. And the next lesson, concluding on this topic, i.e. lesson - test.

Demonstratedslide number 15

The stage of generalization and systematization of knowledge.

1) Task 1.

Compose proportions for solving problems:(work in notebooks)

and)The cyclist covers 75 km in 3 hours. How long will it take for 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 will be able to 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 you get from 54 kg of tomatoes?

Check answers. ( Slide number 16) (self-assessment: put + or - with a pencil innotebooks; analyze errors)

Answers:a) 3: x \u003d 75: 125c) 8: x \u003d 10: 15

b) 8: 10 \u003d X: 2 5 d) 5.6: 54 \u003d 2: X

2) Physical education. (slide number 17-22)

We got up quickly from behind the desks

And they walked on the spot.

And then we smiled

They stretched higher and higher.

Sat down - got up, sat down - got up

We gained strength in a minute.

Straighten your shoulders,

Raise up, down

Right, left turn

And sit down at the desk again.

3) Solve the problem (slide number 23)

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

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

Read the problem.

What two quantities are mentioned in the problem?(about the number of limes and their percentage)

What is the relationship between these values?(directly proportional)

Make a short note, proportion and solve the problem.

Decision:

	Lindens (pcs.)	Interest%
Planted
Accepted

;
; x \u003d 60.

Answer: 60 lindens were planted.

4) Solve the problem: (slide number 24-25) (after parsing, solve it yourself; mutual check, then the solution is displayed on the screen slide number 23)

Coal was prepared for heating the school building for 180 days at a consumption rate of 0.6 tons of coal per day. How many days will this stock last if you spend 0.5 tonnes daily?

Decision:

Brief entry:

Weight (t)

in 1 day

amount

days

At the norm

Let's make the proportion:

;
;
days

Answer: 216 days.

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

(Slide number 26)

In iron ore, 7 parts of iron have 3 parts of impurities. How many tons of impurities are in the ore, which contains 73.5 tons of iron?

Decision: (slide number 27)

amount

parts

Weight

Iron

73,5

Impurities

;
;

Answer: 31.5 kg of impurities.

6) Summing up the results of the final stage. (slide number 28)

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

Algorithm for solving problems in a straight line

and inverse proportional relationships:

The unknown number is indicated by the letter x.

The condition is written in the form of a table.

The type of relationship between values \u200b\u200bis established.

Directly proportional dependence is indicated by equally directed arrows, and inversely proportional dependence is indicated by oppositely directed arrows.

The proportion is recorded.

Its unknown member is located.

5. Repetition of the material studied. (slide number 29)

№763 (s) (p. 125)(with comments at the blackboard)

6. The stage of control and self-control of knowledge and methods of action.
(slide number 30-32)

Independent work (10 - 15 min.) (Mutual check: according to the finished slides, students check each other's independent work, putting + or - at the same time. The teacher at the end of the lesson collects notebooks for viewing).

Solve problems by proportioning.

№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 travel to overcome this path in 0.5 hours?

Decision:

Brief entry:

	Speed \u200b\u200b(km / h)	Time (h)
	12,5

Let's make the proportion:

;
;
km / h

Answer: 17.5 km / h

№2. From 5 kg of fresh plums, 1.5 kg of prunes are obtained. How many prunes do you get from 17.5 kg of fresh plums?

Decision:

Brief entry:

	Plums (kg)	Prunes (kg)

	17,5

Let's make the proportion:

;
;
kg

Answer: 5.25 kg

№3. The car drove 500 km using 35 liters of gasoline. How many liters of gasoline will it take to travel 420 km?

Decision:

Brief entry:

	Distance (km)	Gasoline (l)

The easiest way to understand the direct proportional relationship is on the example of a machine that produces parts at a constant speed. If in two hours he makes 25 parts, then in 4 hours he will make twice as many parts - 50. How many times longer it will work, the same number of times it will make more parts.

Mathematically, it looks like this:

4: 2 = 50: 25 or like this: 2: 4 \u003d 25: 50

The operating time of the machine and the number of parts produced are directly proportional here.

They say: The number of parts is directly proportional to the operating time of the machine.

If two quantities are directly proportional, then the ratios of the corresponding quantities are equal. (In our example, this is the ratio of time 1 to time 2 \u003d the ratio of the number of parts in time 1 to number of parts in time 2)

Inverse proportion

Inversely proportional relationships are common in speed problems. Speed \u200b\u200band time are inversely proportional. Indeed, the faster the object moves, the less time it will take on the way.

For example:

If the quantities are inversely proportional, then the ratio of the values \u200b\u200bof one quantity (speed in our example) is equal to the inverse ratio of another quantity (time in our example). (In our example, the ratio of the first speed to the second speed is equal to the ratio of the second time to the first time.

Examples of tasks

Task 1:

Decision:

Let's write down a short statement of the problem:

Task 2:

Decision:

Brief entry:

If games or simulators are not opening for you, read on.