Class: 6
In my work I use different shapes and teaching methods, I try to use a variety of organizational techniques educational activities so that students are interested in working in class. Only in this case does the cognitive activity of students increase, and thinking begins to work more productively and creatively. One of the means of increasing interest in the subject is the use of information technology.
Usage computer technology in the lesson allows you to continuously change forms of work, constantly alternate oral and written exercises, and implement different approaches to solving mathematical problems, and this constantly creates and maintains the intellectual tension of students, and forms in them a sustainable interest in studying this subject.
Group work in the lesson stimulates the cognitive activity of students, promotes their involvement in creative activities and communication. In the process of individual work, students themselves strive to solve problems; education turns into self-education.
Performance creative tasks promotes the use school knowledge in real life situations.
Lesson type: combined lesson
Lesson objectives:
- Cognitive:
- ensure students’ conscious understanding of the concepts of direct and inverse proportional dependence when solving problems;
- check the level of knowledge on this topic through various forms of work.
- Developmental:
- to activate the mental activity of students through the participation of each of them in the work process;
- develop attention, memory, intellectual and creative abilities;
- develop emotional sphere students in the learning process;
- develop control and self-control.
- Educational:
- to create feelings of cooperation and mutual assistance;
- develop practical skills;
- develop interest in the subject being studied.
Lesson plan:
- Organizational moment (2 min.)
- Oral counting (4 min.)
- Analysis of problems solved by students (5 min.)
- Physical education minute (2 min.)
- Consolidation of the studied material, group work (16 min.)
- Independent work (13 min.)
- Lesson summary (2 min.)
- Homework(1 min.)
DURING THE CLASSES
1. Organizational moment
Mutual greeting, recording the topic of the lesson. Organization of work with self-control cards.
2. Repetition of material
a) Solution of problems involving direct and inverse proportionality by two students on the board
b) the rest orally repeat the basic concepts:
- What are the numbers x and y called in the proportion x: a = b: y?
- equality of two relations is called...
- What kind of relationship is called directly proportional?
- What kind of relationship is called inversely proportional?
- one hundredth of a number is...
Working with self-control cards (maximum number of points – 1).
3. Oral counting
1. Game “Silence”
a) Which of the equalities can be called proportions?
If the proportion is correct, then students raise green cards; if not, then red cards.
b) Are the following relationships directly or inversely proportional?
1) the number of readers from the number of books in the library;
2) the distance traveled by the car at a constant speed and time of its movement;
3) the age of the person and the size of his shoes;
4) the perimeter of the square and the length of its sides;
5) speed and time when passing the same section of the path.
If the statement is true, then students raise green cards; if not, then red cards.
Working with self-control cards (maximum score for oral counting is 2).
2. Analysis of problems solved by students on the board.
a) The swallow flew a certain distance in 0.5 hours at a speed of 50 km/h. How many minutes will it take a swift to fly the same distance if its speed is 100 km/h?
Solution:
Let x hours be the flight time of the swift.
| 50 km/h – 0.5 h 100 km/h – X h |
0.25 h = 25/100 = 1/4 h = 15 min.
Answer: in 15 minutes.
b) Beetroot was brought to the sugar factory, from which 12% sugar is obtained. How much sugar will be produced from 30 tons of beets of this variety?
Solution:
Let x t of sugar turn out.
Answer: 3.6 t.
4. Physical education minute
5. Group work
There are cards on your tables. They have 4 tasks each. Groups 1, 3, 5 decide starting from No. 1. Groups 2, 4, 6 solve starting from number 4 (in reverse order).
1) 80 kg of potatoes contain 14 kg of starch. Find the percentage of starch in such potatoes.
Solution:
Let x% starch be contained in potatoes.
17.5% is starch.
Answer: 17, 5 %
2) You can swim from one village to another along the river in 1.5 hours. How long will it take a motor boat to travel this route if the speed of the boat is 3 km/h and the speed of the boat is 13.5 km/h?
Solution:
Let x hours be the time the boat moves
| 3 km/h 13.5 km/h |
– 1.5 hours – X h |
Answer: 20 minutes
3) When cleaning sunflower seeds, 28% is husk. How much pure grain will be produced from 150 tons of sunflower seeds?
Solution:
Let x t of grain be obtained.
150 – 42 = 108 (t)
108 tons of grain.
Answer: 108 t.
4) To transport the cargo, 48 vehicles with a carrying capacity of 7.5 tons were required. How many vehicles with a carrying capacity of 4.5 tons are needed to transport the same cargo?
Solution:
Let x vehicles be taken with a carrying capacity of 4.5 tons.
Answer: 80 cars.
Checking solutions to problems on the board.
Working with self-control cards (maximum number of points – 8; each task 2 points)
5. Individual independent work 4 options.
Option I
1) Dad paid 48 rubles for 4 identical boxes of pencils. How much do 7 of these boxes of pencils cost?
2) Three students weeded a garden bed in 4 hours. How many hours will it take 2 students to complete the same work?
Option II
1) When cooking meat, 65% of the mass remains. How much cooked meat will you get from 2 kg of raw meat?
2) Four masons can complete the job in 15 days. In how many days can three masons complete this work?
Option III
1) Linden blossom loses 74% of its weight. How much dry linden blossom can be obtained from 300 kg of fresh?
2) A motorcyclist drove for 3 hours at a speed of 60 km/h. How many hours will it take him to cover the same distance at a speed of 45 km/h?
IV option
1) Cuban farmers offer us sugar cane for sugar production. When processed into sugar, sugar cane loses 91% of its original mass. How much sugar cane do you need to get 900 kg of sugar?
2) On a hot day, 6 Kostsy drank a keg of kvass in 1.5 hours. How many Kostsy will drink the same keg in 3 hours?
7. Summing up the lesson
– What types of problems did we solve in class?
Students summarize the lesson in self-control cards and give grades
16-17 points – “5”
13-15 points – “4”
9-12 points – “3”
– The objectives of the lesson were achieved, and most importantly, the work was carried out in a creative atmosphere.
8. Homework
Repeat steps 13-18.
Textbook assignment: No. 817, No. 812, differentiated No. 818.
Literature
- 6th grade mathematics textbook educational institutions, authors: N. Ya. Vilenkin, V. I. Zhokhov, A.S. Chesnokov, S.I. Shvartsburd, Moscow. "Mnemosyne", 2011.
- Collection test tasks for thematic and final control Mathematics 6th grade Moscow, "Intellect-Center" 2009.
- A.I. Ershova, V.V. Goloborodko. Mathematics 6. Independent and test papers.– M: Ilexa, 2011.
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Slide captions:
"Direct and inverse proportional dependencies" 6th grade 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. If you follow her wise laws, you will increase your knowledge and begin to apply them. You can sail 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, try, Learning the salt of science, try to prove everything, But tirelessly.
Finish the phrase: 1. A direct proportional dependence is such a dependence of quantities in which... 2. An inverse proportional dependence is such a dependence of quantities in which... 3. To find the unknown extreme term of the proportion... 4. The middle term of the proportion is equal to... 5. The proportion is correct, if... C) ...as 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 increases 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. U) ...as one value increases several times, the other increases by the same amount. E) ...the ratio of the product of the extreme terms to the known average.
A child's height and age are directly proportional. 2. Given a constant width of a rectangle, its length and area are directly proportional. 3. If the area of a rectangle constant, then its length and width are inversely proportional quantities. 4. The speed of a car and the time it moves are inversely proportional.
5. The speed of a car and its distance traveled are inversely proportional. 6. The revenue of a cinema box office is directly proportional to the number of tickets sold, sold at the same price. 7. The 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 product and its mass are inversely proportional.
Come on, put the pencils aside! No papers, no pens, no chalk! Verbal counting! We do this work only with the power of mind and soul! VERBAL COUNTING
Find the unknown proportion term? ? ? ? ? ? ?
"DIRECT PROPORTIONAL DEPENDENCE" LESSON TOPIC AND REVERSE
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 a pool in 25 minutes. How many minutes will it take to fill a pool with 10 such pipes? c) A team of 8 workers completes the task in 15 days. How many workers can complete this task in 10 days while 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? Create proportions to solve 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 stored for 180 days at a consumption rate of 0.6 tons of coal per day. How many days will this supply last if 0.5t is spent daily? Solve the problem
Brief entry: Mass (t) for 1 day Number of days According to the norm 0.6 180 0.5 x Let's make a proportion: ; ; Answer: 216 days. Solution.
IN iron ore For 7 parts of iron there are 3 parts of impurities. How many tons of impurities are in the ore that contains 73.5 tons of iron? No. 793 Solve the problem
Number of parts Mass Iron 7 73.5 Impurities 3 x; Answer: 31.5 kg of impurities. Solution. ; №793
An unknown number is denoted by the letter x. The condition is written in table form. The type of relationship between quantities is established. Directly proportional dependence is indicated by identically directed arrows, and an inversely proportional relationship is indicated by oppositely directed arrows. The proportion is recorded. Her unknown member is located. Algorithm for solving problems involving direct and inverse proportional relationships:
Solve the equation:
No. 1. The cyclist spent 0.7 hours traveling from one village to another at a speed of 12.5 km/h. At what speed did he have to travel to cover this path in 0.5 hours? No. 2. From 5 kg of fresh plums you get 1.5 kg of prunes. How many prunes will 17.5 kg of fresh plums yield? No. 3. The car traveled 500 km, using 35 liters of gasoline. How many liters of gasoline will be needed to travel 420 km? No. 4. In 2 hours we caught 12 crucian carp. How many crucian carp will be caught in 3 hours? #5 Six painters can complete some work in 18 days. How many more painters must be hired to get the job done in 12 days? Independent work Solve problems by making proportions.
Solutions to 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; ; ;
Solutions to problems from independent work Solution: No. 3 Solution: No. 5 Short entry: Short entry: Distance (km) Gasoline (l) 500 35 420 x; Answer: 29.4 l. Number of malyas 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: 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 of these machines can an enterprise buy if the price for one machine becomes 15 thousand rubles? Solution: No. 1 Brief entry: Number of cars (pieces) 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;
Developmental – through problem solving, posing additional questions and tasks, to develop the creative and mental activity of students;
Independence;
Self-esteem skills;
Educational- cultivate interest in mathematics as part of universal human culture.
Equipment: TSO required for the presentation: computer and projector, sheets of paper for writing down answers, cards for conducting the reflection stage (three for each), pointer.
Lesson type: lesson in applying knowledge.
Forms of lesson organization:frontal, collective, individual work.
During the classes
- Organizing time.
Teacher reads: (slide No. 2)
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
Will you start using them?
Can you 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, try,
Understanding the salt of science.
Try to prove everything
But tirelessly.
2. Checking the studied material.
- Finish the sentence:(slide 3). (Children first complete the task independently, writing down on pieces of paper only the letters corresponding to the correct answer. Then they raise their hands. After that, the teacher reads the question out loud, and the students answer).
- Direct proportional dependence is such a dependence of quantities in which...
- An inverse proportional dependence is a dependence of quantities in which...
- To find the unknown extreme term of the proportion...
- The average term of the proportion is...
- The proportion is correct if...
C) ...as 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 increases 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.
U) ...as one value increases several times, the other increases by the same amount.
E) ...the ratio of the product of the extreme terms to the known average.
Answer: SUCCESS. (slide 6)
- Oral counting: (slides 6-7)
Come on, put the pencils aside!
No papers, no pens, no chalk!
Verbal counting! We're doing this thing
Only by the power of mind and soul!
Exercise: Find the unknown term of the proportion:
Answers: 1) 39; 24; 3; 24; 21.
2)10; 3; 13.
- Lesson topic message. slide number 8 (Provides motivation for schoolchildren to study.)
- The topic of our lesson is “Direct and inverse proportional relationships.”
- In previous lessons, we looked at the direct and inverse proportional dependence of quantities. Today in the lesson we will solve various problems using proportions, establishing the type of connection between data. Let us repeat the basic property of proportions. And the next lesson, concluding on this topic, i.e. lesson - test.
- The stage of generalization and systematization of knowledge.
1) Task1.
Create proportions to solve 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 a pool in 25 minutes. How many minutes will it take to fill a pool with 10 such pipes?
c) A team of 8 workers completes the task in 15 days. How many workers can complete this task in 10 days while 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 No. 10) (self-esteem: put + or – in pencilnotebooks; 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 it yourself)
In the spring, during the city's landscaping work, linden trees were planted on the street. 95% of all planted linden trees were accepted. How many linden trees were planted if 57 linden trees were planted?
- Read the problem.
- What two quantities are discussed in the problem?(about the number of linden trees and their percentages)
- What is the relationship between these quantities?(directly proportional)
- Make a short note, proportion and solve the problem.
Solution:
Linden trees (pcs.) | Interest % |
|
They imprisoned | ||
Accepted |
; ; x=60.
Answer: 60 linden trees were planted.
Solve the problem: (slide No. 11-12) (after analysis, decide on your own; mutual verification, then the solution is displayed on the screen, slide No. 23)
To heat the school building, coal was stored for 180 days at a consumption rate of 0.6 tons of coal per day. How many days will this supply last if 0.5t is spent daily?
Solution:
Brief entry:
Weight (t) in 1 day | Quantity days |
|
According to the norm | ||
Let's make a proportion:
; ; days
Answer: 216 days.
No. 793 (p. 131) (parsing field independently; self-control.
(Slide No. 13)
In iron ore, for every 7 parts iron there are 3 parts impurities. How many tons of impurities are in the ore that contains 73.5 tons of iron?
Solution: (slide No. 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 direct problems
and inversely proportional relationships:
- An unknown number is denoted by the letter x.
- The condition is written in table form.
- The type of relationship between quantities is established.
- A directly proportional relationship is indicated by identically directed arrows, and an inversely proportional relationship is indicated by oppositely directed arrows.
- The proportion is recorded.
- Her unknown member is located.
Repetition of learned material.
No. 763 (i) (p. 125) (with commenting at the board)
6. Stage of control and self-control of knowledge and methods of action.
(slide No. 17-19)
Independent work(10 – 15 min) (Mutual check: students check each other using ready-made slides independent work, while setting + or -. At the end of the lesson, the teacher collects the notebooks for review).
Solve problems by making proportions.
No. 1. The cyclist spent 0.7 hours traveling from one village to another at a speed of 12.5 km/h. At what speed did he have to travel to cover this path in 0.5 hours?
Solution:
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 you get 1.5 kg of prunes. How many prunes will 17.5 kg of fresh plums yield?
Solution:
Brief entry:
Plums (kg) | Prunes (kg) |
|
17,5 |
Let's make a proportion:
; ; kg
Answer: 5.25 kg
No. 3. The car traveled 500 km, using 35 liters of gasoline. How many liters of gasoline will be needed to travel 420 km?
Solution:
Brief entry:
Distance (km) | Gasoline (l) |
|
Let's make a proportion:
; ; l
Answer: 29.4 l.
№4 . In 2 hours we caught 12 crucian carp. How many crucian carp 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 complete some work in 18 days. How many more painters must be hired to get the job done in 12 days?
Solution:
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 No. 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 of these machines can an enterprise buy if the price for one machine becomes 15 thousand rubles?
Solution:
Brief entry:
Number of cars (pcs.) | Price (thousand rubles) |
|
Let's make a proportion:
; ; cars.
Answer: 4 cars.
- Lesson summary stage
- 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 for which the dependence is neither directly nor inversely proportional.
- Homework (slide21)
№ 812, 816, 818.
Thanks for the lesson slide number 22
Mathematics is the basis and queen of all sciences, and I advise you to make friends with it, my friend. If you follow her wise laws, you will increase your knowledge and begin to apply them. You can sail 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, try, Learning the salt of science. Try to prove everything, but without putting your hands on it.
3 Select an answer with the corresponding letter of the hidden word: 17-v; 7-l; 0.1-i; 14-s; 0.2-a; 25-k. Find the missing numbers and find out the word:3+37:5 3. 0.3 +4.1: .45: .7 5.6:0.7:2 0 +4.8:26 word.9 50.050.1 0.050.337 80,45,20,2 s i l a This word is power. Lesson motto: Strength is in knowledge! I'm searching, which means I'm learning!
A direct proportional dependence is such a dependence of quantities in which... An inverse proportional dependence is such a dependence of quantities in 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) ...as 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 increases 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. U) ... when one value increases several times, the other increases by the same amount. E) ... the ratio of the product of the extreme terms to the known average
4. The speed of a car and the time it moves are inversely proportional. 5. The speed of a car and its distance traveled are inversely proportional. 6. Two quantities are called inversely proportional if, when one of them increases by half, the other decreases by half.
Let's check the answers:
Solution. Number of bulldozers Time. (min) x Let's determine the dependence and make up the proportion: 7:5 = 210: x x = 210 * 5: 7 x = 150 (min). 150 min. = 2.5 hours Answer: in 2.5 hours
Algorithm for solving problems involving direct and inverse proportional relationships: An unknown number is denoted by the letter x. The condition is written in table form. The type of relationship between quantities is established. A directly proportional relationship is indicated by identically directed arrows, and an inversely proportional relationship is indicated by oppositely directed arrows. The proportion is recorded. Her unknown member is located.
Test yourself: What quantities are called directly proportional? Give examples of directly proportional quantities. What quantities are called inversely proportional? Give examples of inversely proportional quantities. Give examples of quantities for which the dependence is neither directly nor inversely proportional.
Homework. P; 811; 812.
Proportionality is a relationship between two quantities, in which a change in one of them entails a change in the other by the same amount.
Proportionality can be direct or inverse. In this lesson we will look at each of them.
Lesson contentDirect proportionality
Let's assume that the 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 cover a distance of fifty kilometers.
Let us depict in the figure 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 called directly proportional. And the relationship between such 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 quantity decreases by a certain number of times, then the other decreases by the same number of times.
Let's assume that the original plan was to drive a car 100 km in 2 hours, but after driving 50 km, the driver decided to rest. Then it turns out that by reducing the distance by half, the time will decrease by the same amount. In other words, reducing the distance traveled will lead to a decrease in time by the same amount.
An interesting feature of directly proportional quantities is that their ratio is always constant. That is, when the values of directly proportional quantities change, their ratio remains unchanged.
In the example considered, the distance was initially 50 km and the time was one hour. The ratio of distance to time is the number 50.
But we increased the travel time 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 coefficient of direct proportionality. It shows how much distance there is per hour of movement. In this case, the coefficient plays the role of movement speed, since speed is the ratio of the distance traveled to the time.
Proportions can be made from directly proportional quantities. For example, the ratios make up the proportion:
Fifty kilometers is to one hour as one hundred kilometers is to two hours.
Example 2. The cost and quantity of goods purchased 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. As the cost of a purchased product increases, its quantity increases by the same amount.
Since the cost of a product and its quantity are directly proportional quantities, their ratio is always constant.
Let's write down what is the ratio of thirty rubles to one kilogram
Now let’s write down what the ratio of sixty rubles to two kilograms is. This ratio will again be equal to thirty:
Here the coefficient of direct proportionality is the number 30. This coefficient shows how many rubles are per kilogram of sweets. In this example, the coefficient plays the role of the price of one kilogram of goods, since 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 a motorcyclist's speed was 20 km/h, this means that every hour he covered a distance of 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 motorcyclist's speed was 40 km/h, and he spent 2 hours on the same journey.
It is easy to notice that when the speed changes, the time of movement changes by the same amount. Moreover, it changed in the opposite direction - that is, the speed increased, but the time, on the contrary, decreased.
Quantities such as speed and time are called inversely proportional. And the relationship between such 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 quantity decreases by a certain number of times, then the other increases by the same number of times.
For example, if on the way back the motorcyclist’s speed 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 movement time by the same amount.
The peculiarity of inversely proportional quantities is that their product is always constant. That is, when the values of inversely proportional quantities change, their product remains unchanged.
In the example considered, the distance between cities was 80 km. When the speed and time of movement of the motorcyclist changed, this distance always remained unchanged
A motorcyclist could travel 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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