Direct and inverse proportional relationship. "Direct and inverse proportionality"
Summary of the lesson of mathematics of the teacher of mathematics Trishchenkova N.G.
Class: 6
Topic:"Direct and inverse proportionality"Lesson competition
Lesson location: This lesson is the second in the "Direct and Inverse Proportional" topic and builds on the "Proportions" topic.
Lesson Objectives:
Educational:
- Ensure during the lesson the consolidation of the following basic concepts: proportion, the main property of proportion, directly proportional values, inversely proportional values.
- Improving the skills of solving text problems with the help of proportion. Fixing the main property of proportion on examples of solving equations that have the form of a proportion.
- Continue the formation of learning skills: response planning; self-control skills; verbal counting.
- Control of the degree of assimilation of basic knowledge, skills and abilities on this topic.
Developing:
- Development of skills in applying knowledge in a specific situation.
- Development logical thinking, the ability to highlight the main thing, to generalize, to draw correct logical conclusions.
- Development of skills to compare, correctly formulate tasks and express thoughts.
- Development of independent activity of students.
- Development of cognitive interest.
Educational:
- Education of a healthy lifestyle.
- Formation of scientific outlook, interest in the subject through the content educational material.
- Education of the ability to work in a team, a culture of communication, mutual assistance.
- The education of such qualities of character as perseverance in achieving the goal, the ability not to get lost in problem situations.
Lesson duration: 45 minutes
Lesson type: combined
Lesson structure:
1.Organizing time. Setting goals and objectives of the lesson
2. Actualization of knowledge. oral work
3. Solve problems using proportion
4. Physical education
5. Repetition of the material covered
7. Control testing
8. Homework
9. Summing up the lesson. Grading
The expediency of using a media projector in the lesson:
Intensification of the educational process (increasing the amount of information offered, reducing the time for submitting material);
Improving the efficiency of mastering educational material.
Teaching: according to the textbook N.Ya. Vilenkin "Mathematics 6".
DURING THE CLASSES
Organizing time. Setting goals and objectives of the lesson.
Target: greeting, checking readiness for the lesson, revealing the topic and the overall goal of the lesson, preparing students for work in the lesson and creating a favorable working atmosphere.
Teacher: Hello guys! Now we have a math lesson.
Math, friends
It is impossible not to love.
Very exact science,
Very rigorous science
interesting science -
It's math!
Today we have a lesson in solving problems using proportions.
and we have a lot ahead of us various tasks:
at the beginning of our lesson, we will traditionally conduct oral work, during which we will repeat the theoretical material we need today in the lesson;
we will repeat and bring into the system the techniques we have studied for solving problems using proportions;
we will repeat the ability to use the properties of proportions when solving some types of equations;
let's take a short tour of the history of proportion;
pass a control test, during which you will show your knowledge and skills.
And as the motto of our lesson, I propose to take the words of the wonderful writer S. Ya. Marshak, the author of such famous children's poems as:
"Children in a Cage", "The Tale of stupid little mouse”, “That’s how absent-minded”, etc.
Lesson motto:
“Let every day and every hour
You will get a new one.
May your mind be good
And the heart will be smart.”
Knowledge update. oral work.
Target: preparation of students for the dominant type of educational and cognitive activity.
Teacher: Before we start solving problems, let's turn to oral work, which consists of three tasks.
But in order to successfully cope with task 1, you must answer the following questions:
What is a proportion? Students' responses.
Formulate the main property of proportion. Students' responses.
Teacher: Getting to task 1
Exercise 1. Name the extreme and middle members of the proportion:
Answer: Extreme terms 5 and 12, middle terms 10 and 6
Answer: Extreme terms 20 and 7, middle terms 4 and 35
Teacher: You, well done! In order to proceed to the second task, we need to remember the answers to questions such as:
1. What proportion is called correct? Students' responses.
2. What methods help to determine if the proportion is correct? Students' responses.
Teacher: Getting to task 2
Task 2. Indicate the correct proportion:
a) 2:3 = 5:10 Answer: not true
b) 5:10 = 8:4 Answer: not correct
c) 2:3 = 10:15 Answer: correct
d) 3:5 = 10:12 Answer: not true
e) 16:6 = 8:3 Answer: correct
Teacher: You were on top again! The last task remains.
In our port there are three ships "Victory", "Dream" and "Slava" and three piers: A, B, C. It is necessary to put each ship on its own pier, and for this, make the correct proportions from these relations
Task 3. Find a pier for the ship
Piers:
Ships:
Pobeda 105:21
Dream 2:0.5
Slava 6:0.2
Student responses:
90: 3 \u003d 6: 0.2 (A "Glory");
64: 16 \u003d 2: 0.5 (In "Dream");
0.15: 0.03 \u003d 105: 21 (With Pobeda)
Solving problems using proportions.
Target: systematize the studied methods of solving problems using proportions
Preparatory work
Teacher: Guys, today in the lesson we continue to solve problems for direct and inverse proportionality. And in order to cope with the tasks, let's remember:
What quantities are directly proportional?
What quantities are called inversely proportional?
Give examples of directly and inversely proportional quantities.
How can you solve direct and inverse proportionality problems?
What needs to be done to solve the problem using proportions?
Teacher: Let's remember the algorithm for solving proportion problems.
Student responses:
2. An unknown number is denoted by the letter X.
3. Write down the condition of the problem in the form of a table.
4. Determine the type of dependence.
5. Put the arrows corresponding to the type of proportion.
6. Write down the proportion.
7. Find an unknown member of the proportion.
Frontal teamwork
Teacher: Guys, open your notebooks. Now we will start solving problems.
And what will be our first task, we will find out with you by guessing the riddle.
Under the bushes
Under the sheets
We hid in the grass
Look for us in the forest yourself,
We will not shout to you: "Ay!"
Answer: Mushrooms
Task #1
A squirrel from 30 kg of fresh mushrooms received 9 kg of dried ones.
How many fresh mushrooms does he need to collect in the forest to get 15 kg of dried ones? (Answer: 50 kg)
Teacher: Guys, tell me what edible and inedible mushrooms do you know? Students' responses.
Teacher: Let's move on to the second task.
Task #2
3 janitors can sweep an area in 7 hours.
How long will it take the janitors to sweep the same area if 4 more janitors come to their aid? (Answer: 3 hours)
Note: When solving problems, the teacher asks questions:
Briefly describe the problem.
What is known about the problem?
What do you need to know?
What is the relationship between...?
Explain why?
How is this ... dependence indicated on the drawing?
Which term of the proportion is unknown?
How to find an unknown ... term of a proportion?
Work in pairs
Teacher: Guys, and now I suggest you work on tasks in pairs. Pairs are formed in accordance with how you sit at your desks in the lesson.
Now, I will give each pair a card with a picture of a dwarf or a fairy. In accordance with what is shown on your card, you solve a problem in which your character is the main character.
After you solve the problems, we will check the correctness of your solutions.
Note: cards are dealt with a differentiated approach, since inverse proportionality problems cause difficulty.
Problem about gnomes(The problem of direct proportionality)
4 gnomes planted 8 rose bushes for Snow White.
How many rose bushes will 3 gnomes plant in the same time? (Answer: 6 bushes)
Fairy problem(Inverse proportionality problem)
3 fairies will collect honey from flowers in 4 hours.
How many hours will it take 2 fairies to do this work? (Answer: 6 hours)
Note: Students work on tasks. Checking the work done through a slide show on the screen.
Physical education minute
Target: relieve fatigue in students, provide active recreation and increase mental performance.
Teacher: Guys, you are great! You all did a great job, and it's time to relax and spend a physical education minute.
We stomp our feet
We clap our hands
We nod our heads.
We raise our hands
We lower our hands
And let's start writing again.
Repetition of the material covered.
Equations.
Target: consolidate the skills of solving equations written in the form of a proportion.
Teacher: In previous lessons, we talked about , that with the help of proportion it is possible to solve not only problems for direct and inverse proportionality, but also equations.
The dwarves from the fairy tale about Snow White prepared this task for us. Some of you have already helped them plant roses today, and now let's all together and together help them with solving equations.
Let's remember how equations are solved of this type.
Note: Two students are called in turn to the board to work on solving equations. The rest of the students work in notebooks.
During the assignment, the teacher conducts a conversation on the following questions:
Which term of the proportion is unknown? Students' responses.
How to find the unknown extreme term of the proportion? Students' responses.
How to check if you have solved the equation correctly? Students' responses.
Equation 1
( Answer: x = 6)
Equation 2
(Answer: y = 28)
V. Historical reference.
Target: deepening and expanding knowledge of proportion.
Teacher: The world of proportion is vast and varied.
Proportions have been studied since ancient times.
The word "proportion" was introduced by Cicero (an ancient Roman politician and philosopher) in the 1st century BC.
In the 4th century BC. The ancient Greek mathematician Eudoxus gave the definition of proportion.
The history of recording proportions is very interesting.
In 1631, William Outred (English mathematician. Known as the inventor of the slide rule) proposed the following record for the proportion a ● b:: c ● d
Rene Descartes (French mathematician, philosopher, physicist and physiologist. Descartes first introduced the coordinate system.) In the 17th century, he wrote down the proportion as follows:
7 | 12 | 84 | 144 .
In 1693, G. W. Leibniz (German philosopher, logician, mathematician,
physicist, lawyer, historian, diplomat, inventor and linguist) proposed a modern notation for the proportion a: b = c: d.
Portrait of Luca Pacioli
predp. Jacopo de Barbari, 1495
Pacioli was born around 1445 in the small town of Borgo San Sepolcro on the border of Tuscany and Umbria.
As a teenager, he was sent to study in the workshop of the famous artist Piero della Francesca. Here he was noticed by the great Italian architect Leon Batista Alberti, who in 1464 recommended the young man to the wealthy Venetian merchant Antonio de Rompiasi as a home teacher. In 1494 Pacioli published on Italian mathematical work entitled "The sum of arithmetic, geometry, fractions, proportions and proportionality" (Summa di arithmetica, geometrica, proportione et proportionalita), dedicated to the Duke of Urbino, Guidobaldo da Montefeltro. This essay outlines the rules and techniques for arithmetic operations on integer and fractional numbers, proportions, compound interest problems, solving linear, quadratic and certain types of biquadratic equations. It is noteworthy that the book was written not in the usual Latin for scholarly works, but in Italian.
Homework.
Target: give homework that would enable students to realize themselves creatively, to apply the acquired knowledge in a new situation.
Teacher: And your homework will be unusual, creative. It is necessary to come up with an interesting text problem, which is solved with the help of proportions and colorfully arrange it on a landscape sheet.
VIII. Summing up the lesson. Grading.
Target: evaluate the work of students in the classroom.
Teacher: Guys, let's sum up our lesson. Please answer the questions:
What new did you learn in today's lesson, what did you repeat? Students' responses.
What was interesting or not interesting about the lesson? Students' responses.
Guys, thank you for your hard work! You are all great!
The easiest way to understand a directly proportional relationship is to use the example of a machine that manufactures 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 the time he will work, the same number of times more details he will produce.
Mathematically it looks like this:
4: 2 = 50: 25 or like this: 2:4 = 25:50
Directly proportional quantities here are the operating time of the machine and the number of manufactured parts.
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 = the ratio of the number of parts in time 1 To number of parts in time 2)
Inverse proportionality
An inversely proportional relationship is often found in speed problems. Speed and time are inversely proportional. Indeed, the faster an object moves, the less time it will take to travel.
For instance: 
If the quantities are inversely proportional, then the ratio of the values of one quantity (speed in our example) is equal to the inverse ratio of the other 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.
Task examples
Task 1:
Solution:
Let's write a brief condition of the problem: 
Task 2:
Solution:
Brief entry:
Back forward
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Subject: mathematics; Grade 6 (textbook "Mathematics 6" N.Ya. Vilenkin and others)
Topic: Direct and inverse proportions.
Lesson type: learning new material using information technology
Targets and goals:
- Educational:
- to consolidate the basic concepts: proportion, the main property of proportion;
- to form in students the concepts of direct and inverse proportionality;
- to form the ability to solve problems using proportions;
- Educational:
- think logically when determining dependency in accordance with the condition of the problem;
- develop competent mathematical speech; memory, attention, draw conclusions based on reasoning;
- to promote the development of cognitive interest, creativity, the ability to compare, analyze;
- Educational:
- instill an interest in mathematics;
- develop mindfulness skills.
Teaching methods: communicative, differentiated, research and search.
Lesson organization forms: frontal survey, individual work, self-examination.
Equipment: m/m projector, screen, computer, monitor, presentation.
slide number |
Note |
|
| 1 | Organizing time | All slides change on mouse click |
| 2-3 | Knowledge update | Recall the basic concepts: proportion, the main property of proportion (frontal survey) |
| 4 | Oral discussion of ways to solve problems of a new type (search for a solution) | In the course of oral condemnation, determine how dependent quantities change. |
| 5-8 | Test yourself - test work | The theoretical test allows you to adjust the further supply of material |
| 9-10 | Peer-check using m/m projector | Work in pairs of shifts |
| Solving problems on the topic of the lesson (study of solving problems of a new type for proportional dependence) | Work with a textbook, individual work - a differentiated approach | |
| 11-12 | Direct proportional dependence | № 784 |
| 13-14 | № 785 | |
| 15-16 | Inverse Proportional | № 836 |
| 17 | Relaxation, debriefing | |
| 18 | Homework | item 22, No. 805; 811; 812 |
DURING THE CLASSES
1. Organizational stage
Greetings;
Checking students' readiness for the lesson.
– Today we will get acquainted with new concepts: direct and inverse proportional relationships, and we will learn to solve problems based on new knowledge.
2. Updating the basic knowledge and skills of students(slide 2)
- What is a proportion?
- Formulate the main property of proportion.
- What permutations of the terms of the proportion again lead to the correct proportions?
- Make three new correct proportions from the proportion: 5: 15 \u003d 4: 12
- What permutations of the terms of this proportion again lead to the correct proportions?
- Make three new correct proportions from the proportion: (slide 3)
a) 135: __ = 90: 2
b) 18:3 = __ : __
Which of these tasks has a single solution, and which one has many solutions? Why?
Statement of an educational problem for students
– Will the acquired knowledge help us in solving practical problems?
3. Formation of new knowledge
Oral discussion (search for a solution) (slide 4)
1. For 2 kg of vegetables they paid 10 rubles. How much do 8 kg of vegetables cost?
- How many times more vegetables did you buy?
- If you buy more, do you have to pay less or more?
Conclusion: if the quantity of goods increases several times, then the purchase price increases by the same amount.
In the course of verbal condemnation, students determine how dependent quantities change in a given task.
Definition: two quantities are said to be directly proportional if, when one of them increases (decreases) several times, the other increases (decreases) by the same amount.
2. Two tractors plowed a field in 6 days. In how many days will 4 tractors plow this field if they work with the same productivity?
- If there are more tractors, will it take more or less days to plow the same field?
- How many times has the number of tractors increased? How many times fewer days will it take to do the same job?
In the course of verbal condemnation, students determine how dependent quantities change in this task.
Definition: two quantities are said to be inversely proportional if, when one of them increases (decreases) several times, the other decreases (increases) by the same amount
Test work - test yourself
The theoretical test allows you to adjust the further presentation of the material (slides 6; 7; 8)
“Yes” and “no” do not say, depict them with a sign: (slide 5)
"Yes"- sign «+»
,
"No"- sign «–»
.
- The relationship between the quantity of goods and the purchase price is directly proportional.
- The growth of the child and his age are directly proportional.
- With a constant width of a rectangle, its length and area are directly proportional.
- The speed of the car and the time of its movement are inversely proportional.
- The speed of a car and its distance traveled are inversely proportional.
- Two quantities are said to be inversely proportional if when one of them is doubled, the other is doubled.
- The load capacity of the machines and their number are directly proportional.
- The perimeter of a square and the length of its side are directly proportional.
Let's check the answers: mutual check using m / m projector (slide 9): + – + + – + – +
Rate yourself:(slide 10)
8 correct answers - "5"
7-6 correct answers - "4"
5-4 correct answers - "3"
4. Physical education
5. Formation of skills and abilities
Solving the problems of the level of compulsory training (slides 11; 12)
6. Initial check stage
Students perform independent work according to options with mutual verification in pairs.
1 option - No. 785;
Option 2 - No. 836;
Checking the solution: Option 1 - slide 14; Option 2 - slide 16)
7. Summing up the lesson. Reflection
Test yourself:(slide 17)
- What quantities are directly proportional? Give examples of directly proportional quantities.
- What quantities are called inversely proportional? Give examples of inversely proportional quantities.
- Give examples of quantities whose dependence is neither directly nor inversely proportional.
8. Setting homework(slide 18)
- study item 22, No. 805; 811; 812;
- compose the text of two tasks for direct and inverse proportional dependencies (the solution in the next lesson will be performed by a neighbor on the desk).
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.
