“Finding a number from its fraction” - Mathematics textbook, grade 6 (Vilenkin)
Short description:
You already know how to find a fraction from a number, and in this section you will learn how to find a number from its fraction. You need to be very careful not to get confused, and solve all problems quickly and correctly.
Let's quickly remember how we find a fraction from a number: we simply multiply this number by the fraction. For example, you need to find 3/5 of the number 15. Solve 3/5 * 15 = 3*15 / 5 =3*3=9. Why do we need to know how to do this? In order to be able to find some part of something whole. For example, knowing which part of the book you have read and how many pages there are in total, you can find out how many pages are left to read. Remember, when we are looking for a fraction of a number, we have something whole and its part, and we need to multiply this whole by the part, thus we find the part in quantitative terms and this number will always be less than the initial number.
In problems when we are looking for a number by its fraction, this number should always be larger, because, in fact, we are looking for something whole, knowing only its part. For example, you have read 100 pages of a book, but this is only its third part. How many pages are in the book? How will we look for this number? Knowing that 100 pages is a third, we need 100 * 3 and then we will find out how many pages there are in the book - 100 * 3 = 300. What if you try to solve through an equation? Let x be the total number of pages in the book, how to find how many we have read, you need to multiply x by 1/3 and it will be equal to 100. So – x * 1/3=100. We solve the equation further - x = 100: 1/3, and we have already learned that in order to divide a number by a fraction, you need to multiply it by the reciprocal fraction. It turns out x=100: 1/3 = 100 * 3/1 = 300. Got it? This means that in order to find a number, knowing its fractional part and its value, we need to divide the value (natural number) by a fraction, that is, multiply by an inverted fraction and this number will always be greater than the one given to us in the condition!
If the problem gives not a fraction, but a percentage, what should you do? Convert percentages to decimal fractions: 40%=0.40; 75% = 0.75 and solve further according to the learned scheme.
Percent is one hundredth of a number. It follows that two percent is two hundredths, twenty percent is twenty hundredths, and so on.
The word percentage is denoted by the % sign. So, 43% of a number means 43 percent, that is, of that number. However, it is worth noting that the % sign is not written in calculations; it can be written in the problem statement and in the final result.
The value from which percentages are calculated (for example, price, length, number of candies, etc.) is 100 of its hundredths, that is, 100%.
To find one percent of a number, you divide that number by 100.
Example 1. Find one percent of the number 300.
Solution:
Answer: One percent of 300 is equal to 3.
Example 2. Find one percent of the number 27.5
Solution:
27,5: 100 = 0,275
Answer: One percent of 27.5 is equal to 0.275.
Finding percentages of a number
To find a certain percentage of a given number, you need to divide the given number by 100 and multiply by the number of percentages.
Task 1. That year, the store bought 200 Christmas trees for the New Year. This year the number of purchased Christmas trees has increased by 120%. How many Christmas trees did you buy this year?
Solution: First we need to find 120% of 200, for this we need to divide 200 by 100, so we find 1%, and then multiply the result by 120:
(200: 100) 120 = 240
The number 240 is 120% of 200. This means that this year the number of Christmas trees sold increased by 240 pieces. That is, the number of Christmas trees sold this year is equal to:
200 + 240 = 440 (trees)
Answer: This year we bought 440 Christmas trees.
Task 2. There are 28 candies in a box, 25% of candies with strawberry filling. How many candies with strawberry filling are in the box?
Solution:
Answer: The box contains 7 candies with strawberry filling.
Finding a number by its percentage
To find a number from a given percentage, you need to divide this value by the number of percentages and multiply by 100.
Task. The price of a meter of cloth decreased by 24 rubles, which was 15% of the price. How much did a meter of cloth cost before the reduction?
Solution:
Answer: A meter of cloth cost 160 rubles.
Percentage of two numbers
To find out what percentage the first number is of the second, you need to divide the first number by the second and multiply the result by 100.
Task. According to the annual plan, the plant must produce products worth 1,250,000 rubles. During the 1st quarter, he issued it in the amount of 450,000 rubles. By what percentage did the plant fulfill its annual plan for the 1st quarter?
Solution:
Answer: For the 1st quarter the plan was fulfilled by 36%.
Converting percentages to decimals
To convert percentages to decimals, divide the percentage by 100.
Example 1: Express 25% as a decimal.
Answer: 25% is 0.25.
Example 2: Express 100% as a decimal.
Answer: 100% is 1.
Example 3: Express 230% as a decimal.
Answer: 230% is 2.3.
From these examples it follows that To convert percentages to decimal fractions, you need to move the decimal point two places to the left in the number before the % sign..
Math lesson.
Class: 6
Topic: “Finding numbers by their fractions.”
Lesson objectives:
Educational:
Developmental:
Educational:
nurturing interest in the subject through the use of multimedia capabilities of a computer;
Lesson type: combined lesson.
Equipment: screen, PC, projector, presentation, cards, textbook.
Plan:
Organizing time
Checking homework.
Verbal counting
Learning new material
Test
Lesson summary
Homework
Reflection
During the classes
1. Organizational moment
–Hello guys! Today we have guests at our lesson, let's greet them and say hello! Have a seat. I am very glad to see you today. My name is Tatyana Mikhailovna.
2. Checking homework
- Please tell me what was assigned to you at home?
(No. 635 (d,f), No. 641)
- Please look at the slide where the homework problem has been solved and compare with your solution
Total – 156 notebooks
I- ? notebooks
II- ? notebooks - this is from
Solution:
Let there be x notebooks in 1 pack, then x notebooks in 2 pack
x =156;
x = 156: ;
x = 156: 
;
x = 156* 
;
x = 84. (tet.) - in 1 pack
Answer: 84 notebooks, 72 notebooks.
- Well done!
- Today I would like to start the lesson with the following statement: “Consider unhappy that day or that hour in which you have not learned anything new and have not added anything to your education.” (Y.-A. Kamen skiy)
- These words will be the motto of our lesson. And this day will not be unhappy, because we will again learn something new, We will strengthen the skills of finding a fraction from a number, multiplying and dividing ordinary fractions, converting % to decimals and vice versa.
- Guys, tell me, what month started?
(December)
- What time of year is December?
(winter)
- What is the most long-awaited holiday in winter?
(New Year)
We always prepare for this friendly and cheerful holiday, buy gifts, decorate the place where we live and spend a lot of time, and also decorate the Christmas tree.
And today in class I invite you to take part in a small project “Our New Year Tree”. This will not be the project itself, but preparation for it, because the tree is part of the New Year holiday.
2. Oral counting
First, I suggest you light the garland for our Christmas tree!
Let's start the New Year's mental counting! In front of you is a New Year's garland, if you count or answer correctly, its lights will become multi-colored.
Next task:
How to multiply two ordinary fractions?
How to divide by a common fraction?
What numbers are called reciprocals?
Guys, how to convert % into a number?
(% divided by 100)
How do you convert a number to a percentage?
(multiply the number by 100)
And so the next task (Slide)
0,65 65%
0,3 30%
48% 0,48
150% 1,5
Who can tell me how to find a fraction of a number?
(To find a fraction of a number you need to multiply this number by this fraction)
from 36; 28
0.4 from 60; 24
1.2 from 0.5; 0.6
Next task:
There are 60 balls on the Christmas tree. of which are red. How many red balls?
(10)
Well done guys, Val and I decorated our New Year tree with a garland.
Explanation of new material
Guys. And what do they decorate the Christmas tree with after the garland?
(star)
And so the next task is “New Year’s Star”
Please read the task on the slide
« The snow was cleared from the skating rink, which is 800 m 2 . Find the area of the entire skating rink.
- What is known in the problem?
(cleared, and this is 800 m 2 )
- A 800 m 2 Is this part of the skating rink or the entire skating rink?
(Part)
_What do you need to find in the problem?
(Area of the entire skating rink)
- Let x m 2 the whole skating rink
Once you've cleared the snow, how do you find a fraction of a number?
(You need to multiply this number by this fraction)
THOSE. X *
- Do we know what this is equal to?
(800)
- Let's make an equation
X * = 800
What is the main action
(Multiplication)
- name the components
(1 factor, 2 factor, product)
- what is unknown?
(1 multiplier)
- how will we find it?
(1 factor = product: by 2 factor)
X = 800:
X = 800 *
X = 1600 m 2
And so the area of the entire skating rink is 1600 m 2
Guys, in the problem we didn’t know the number itself, but we knew what it was equal to. those are part of it, that is, using its fraction we found the number itself.
So let's concludeTo find a number by its fraction, you need to divide this number by this fraction.
Children, everything is elementary!
I’ll explain it popularly:
You don't need to be a genius here,
And the number given to us
Let's start dividing by fractions.
And so guys, we were able to decorate our Christmas tree with a New Year's star.
Fizminutka
The music plays and the child comes out and does some physical exercise.
Together we counted and talked about numbers,
And now we stood up together and stretched our bones.
On the count of one we clench our fist, on the count of two we clench our elbows.
On the count of three, press it to your shoulders, on 4, press it to the heavens.
We bent over well and smiled at each other
Let's not forget about the top five - we will always be kind.
On the count of six, I ask everyone to sit down.
Numbers, me and you, friends, together are the friendly 7th.
4. Consolidation of learned knowledge.
Well, you have completed all my previous tasks, so I suggest moving on to the next stage of decorating the Christmas tree “New Year’s ball”. – At this stage we will solve problems on finding a number by its fraction and decorate the Christmas tree with New Year’s toys.
Guys, please look at the board, there are examples written on the board that you and I must solve
(for each example, 1 student hangs balls after solving the solution)
Find the number if:
of this number are 24 = 56
0.6 of this number equals 6 = 10
0.3 of this number equals 33 = 110
Guys, please look at the slide.
3) Guys, on your tables there are worksheets with which we will solve more than one problem today. So, read carefully the conditions of task No. 1 and pay attention to what we know in the problem and what needs to be found.
Total - ? km
By car – 30 km
Solution:
Answer: 50 km
Total - ? games.
6th grade – 15 games. - This
Other classes - ? games.
Solution:
Answer: 30 toys
After solving two problems, 3 students solve the test on the computer, and the rest continue to solve the problems.
Independent work
K)49; L)64; M)56.
E)90; G)10; Z)20.
B)30; D)4; D)25.
Answers:
1
Total - ? gir.
6th grade – 3rd weight. - This
The rest of the students - ? gir.
Solution:
1)3: = 11 (weights) – total
2) 11-3 = 8 (weight) – other classes
Answer: 8 garlands
Total - ? windows
I – 30 windows – that’s
II- ? windows
Solution:
30: 0,6 = 50 (windows) - total in school
50 – 30 = 20 (windows) – on day 2
Answer: 20 windows
Lesson summary
– Our lesson is coming to an end, let's summarize it.
WHAT RULES WE REPEATED IN TODAY'S LESSON?
What rule did we meet today?
And so if you look, we started preparing for the New Year, brought and decorated the Christmas tree, and in all this we were helped by our favorite mathematics and our topic “Finding numbers by their fractions”
For homework, I offer you the tasks PRESENTED IN YOUR WORKSHEETS.
Homework.
3. Mom asked her son to water 0.2 from all the flower beds at the dacha. My son quickly calculated and said that it wouldn’t be difficult for me to water one flowerbed well. How many flower beds are there in the country house?
4. Five friends bought candy and ate three pieces at once, this amounted to
At the end of our lesson we must complete The most enjoyable task is to dress up our green beauty colorful balls! These SMILE balls are lying on your tables, choose the one that matches your mood and, when you leave, attach it to our Christmas tree!
Those guys who received gifts can submit diaries for grading.
THANK YOU ALL SO MUCH FOR THE LESSON! I wish you good luck in the next lessons.
The red card means: “I am satisfied with the lesson, the lesson was useful for me, I worked a lot, usefully and well in the lesson, I understood everything that was said and what was done in the lesson.”
A yellow card means: “The lesson was interesting, I took an active part in it, the lesson was useful to me to a certain extent, I answered from my seat, I was able to complete a number of tasks, I felt quite comfortable in the lesson.”
The blue card means: “I got little benefit from the lesson, I didn’t really understand what was going on, I don’t really need it, I won’t do my homework, I’m not interested in it, I wasn’t ready for the answers in the lesson.” .
WORKSHEET
systematize students’ knowledge about dividing ordinary fractions;
practice skills in performing operations with ordinary fractions;
contribute to the formation of the ability to solve problems of finding a number by its part, expressed as a fraction, by dividing by a fraction;
create organizational conditions for the development of students’ ability to analyze and compare;
create positive motivation in students to perform mental and practical actions, promote the development of the ability to cooperate.
The schoolchildren spent two days decorating the windows at the school. On the first day
0.6 of all windows were taken, which amounted to 30 windows. How many windows were decorated on the second day?
Homework.
1. Find the value of the quantity if:
a) 0.8 of it is equal to 576 g; b) 2/9 of it are equal to 36l;
c) 24% of it is equal to 57.6 km; d) 2.3% of it is equal to 2.07 rubles.
2. For a gift for the boy, friends collected one-fourth of the cost of the bicycle, which amounted to 120 rubles. How much money do the guys need to buy a gift?
1. Mom asked her son to water 0.2 from all the flower beds at the dacha. My son quickly calculated and said that it wouldn’t be difficult for me to water one flowerbed well. How many flower beds are there in the country house?2. Five friends bought candy and ate three pieces at once, this amounted to the total amount. How many total candies were purchased?Introspection.
Subject: " Finding a number from its part ».
Lesson objectives:
Educational:
Developmental:
promote the development of logical thinking and memory;
develop the ability to analyze the situation and evaluate the results of activities;
develop independence and attention.
Educational:
nurturing interest in the subject through the use of multimedia capabilities of the computer, as well as interest in New Year traditions.
fostering accuracy when preparing work.
The objectives of the lesson are aimed at knowledge and skills:
Understand the educational task, carry out the solution of the educational task both under the guidance of the teacher and independently, control your actions in the process of its implementation, detect and correct mistakes, both other people’s and your own, evaluate your achievements.
To cultivate a love of mathematics, interest in it, respect for each other, listening skills, discipline, and independence.
F develop skills in dividing and multiplying ordinary fractions, correctly read and write expressions containing ordinary fractions, develop the ability to solve problems on the topic “Finding a number from its fraction.”
Lesson type: learning new material.
Equipment: screen, PC, projector, presentation, worksheets.
Forms lesson organization:
Frontal
individual
Teaching methods:
Visual
Problem-search
Reproductive
Description of the lesson
The topic of the lesson is reflected in thematic planning and represents 1 lesson out of 5 in the topic “Finding a number by its part” and is based on the content of three topics: “Reciprocal numbers”, “Multiplying fractions” and “Dividing fractions”. I wanted the students in this lesson to see the connection between this topic and what they had previously studied and to realize(which is especially important in mathematics) that all topics are closely interconnected and cannot be studied in isolation from each other. During the lesson, the children apply the knowledge gained not only in this lesson, but also in previous lessons.
The structure of the lesson consisted of 9 main stages
Organizing time
Checking homework.
Verbal counting
Learning new material
Reinforcing the material learned
Test
Lesson summary
Homework
Reflection
At the beginning of the lesson, org. moment allowed me to tune in to the lesson. Allowed us to give a positive attitude towards fruitful cooperation.
Onstage of oral counting The goal was to include students in work, determine the scope of work in the lesson, and set a goal for students: creating a game situation regarding the project “Our New Year Tree.” Oral work in a game form made it possible to create a situation of success and corresponded to the psychological characteristics of the age. Mathematical dictation contributed developing the ability to correctly read expressions containing ordinary fractions, as well as perform actions independently and evaluate one’s achievements.
At the stage learning new materialThe children were asked to come to the conclusion for themselves thatto find a number by its fraction you need this number ra divide by this fraction.
At the consolidation stagestudied material frontal and individual work were used, skills of dividing and multiplying ordinary fractions were formed. Self-examination (test) contributed to the formation of the ability to see one’s mistakes and evaluate one’s achievements.
Homework explanation stage helped to arouse students' interest. The assignments are practice-oriented in nature and help to convince the children that mathematics is a science closely related to life.
Reflection stage became a logical conclusion to the lesson and helped students express their attitude to the lesson, and helped me, as a teacher, see the assessment of my lesson.
Thus, the goals set for the lesson, in my opinion, were achieved.
In this lesson we will look at the types of problems involving fractions and percentages. Let's learn how to solve these problems and find out which of them we may encounter in real life. Let's find out a general algorithm for solving similar problems.
We don’t know what the original number was, but we know how much it turned out when we took a certain fraction from it. We need to find the original.
That is, we don’t know, but we also know.
Example 4
Grandfather spent his life in the village, which was 63 years. How old is grandpa?
We do not know the original number - age. But we know the share and how many years this share is from the age. We make up an equality. It has the form of an equation with an unknown. We express and find it.
Answer: 84 years old.
Not a very realistic task. It is unlikely that grandfather will give out such information about his years of life.
But the following situation is very common.
Example 5
5% discount in the store using the card. The buyer received a discount of 30 rubles. What was the purchase price before the discount?
We do not know the original number - the purchase price. But we know the fraction (the percentages that are written on the card) and how much the discount was.
Let's create our standard line. We express the unknown quantity and find it.
Answer: 600 rubles.
Example 6
We are faced with this problem even more often. We see not the amount of the discount, but what the cost is after applying the discount. But the question is the same: how much would we pay without the discount?
Let us again have a 5% discount card. We showed our card at the checkout and paid 1,140 rubles. What is the cost without discount?
To solve the problem in one step, let’s reformulate it a little. Since we have a 5% discount, how much do we pay from the full price? 95%.
That is, we do not know the original cost, but we know that 95% of it is 1140 rubles.
We apply the algorithm. We get the initial cost.
3. Website “Mathematics Online” ()
Homework
1. Mathematics. 6th grade/N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwartzburd. - M.: Mnemosyne, 2011. Pp. 104-105. clause 18. No. 680; No. 683; No. 783 (a, b)
2. Mathematics. 6th grade/N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwartzburd. - M.: Mnemosyne, 2011. No. 656.
3. The program of school sports competitions included long jump, high jump and running. All participants took part in the running competition, 30% of all participants took part in the long jump competition, and the remaining 34 students took part in the high jump competition. Find the number of participants in the competition.