The rule for finding a number by its fraction:
To find a number from a given value of its fraction, you need to divide this value by the fraction.
Let's look at how to find a number by its fraction, using specific examples.
Examples.
1) Find a number whose 3/4 are equal to 12.
To find a number by its fraction, divide the number by that fraction. To do this, you need to multiply this number by the inverse of the fraction (that is, by an inverted fraction). To do this, you need to multiply the numerator by this number and leave the denominator unchanged. 12 and 3 by 3. Since we got one in the denominator, the answer is an integer.
2) Find a number if 9/10 of it equals 3/5.
To find a number given the value of its fraction, divide this value by this fraction. To divide a fraction by a fraction, multiply the first fraction by the inverse of the second (inverted). To multiply a fraction by a fraction, multiply the numerator by the numerator, and the denominator by the denominator. We reduce 10 and 5 by 5, 3 and 9 by 3. As a result, we get the correct irreducible fraction, which means this is the final result.
3) Find a number whose 9/7 are equal
To find a number by the value of its fraction, divide that value by that fraction. Mixed number and multiply it by the inverse of the second number (inverted fraction). We reduce 99 and 9 by 9, 7 and 14 by 7. Since we received an improper fraction, we need to separate the whole part from it.
Class: 6
Presentations for the lesson
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Epigraph for the lesson:
“The one who learns on his own succeeds seven times more than the one to whom everything is explained” (Arthur Giterman, German poet)
Lesson type: lesson on learning new material.
Methods: partial search.
Forms: individual, collective, group, individual.
(Place - 1 lesson on the topic)
Type of lesson: explanatory and illustrative
The purpose of the lesson: to come up with a new way to solve problems on fractions, to strengthen the skills and abilities of solving problems.
- systematize the solution of problems into parts, develop a new technique for solving problems of finding a number from its part.
- help develop students’ interest not only in the content, but also in the process of acquiring knowledge, and expand the mental horizons of students. Development of students' thinking, mathematical speech, motivational sphere of personality, research skills.
- to instill in students a sense of satisfaction from the opportunity to show their knowledge in class. To create positive motivation among schoolchildren to perform mental and practical actions. Fostering responsibility, organization, and perseverance in solving tasks.
Equipment: illustrative material, presentation for the lesson. Sheets with tasks for reflection, mathematics textbook Mathematics. 6th grade / N. Ya. Vilenkin, V. I. Zhokhov, A.S. Chesnokov, S. I. Shvartsburd. M.: Mnemosyne, 2011.
Lesson plan:
- Organizing time.
During the classes
1. Organizational moment.
(Didactic task – psychological mood of students)
Hello, please sit down. We inform the topic, the goals of the lesson and the practical significance of the topic.
The goal of our lesson is to come up with a new way to solve fraction problems.
2. Updating basic knowledge and correcting it
(The didactic task is to prepare students for work in class. Ensuring students’ motivation and acceptance of goals, educational and cognitive activities, updating basic knowledge and skills).
15; ; 3 6; ; (2; ; 19; c)
Questions for the class:
– How to multiply a fraction by a natural number?
– How to find the product of fractions?
– How to find the product of a mixed number and a number? (using the distributive property of multiplication or converting a mixed number to an improper fraction)
– How to multiply mixed numbers?
2) :2; V:; :; :; (; ; ; X)
Questions for the class:
– How to divide a fraction by a natural number?
– How to divide one fraction by another?
– How to divide a mixed number by a mixed number?
Tables on the slide and supports on the desks of the weak group:
Repeat algorithms for solving problems of finding a number by its part.
1) We cleared the snow from the skating rink, which is 800 m2. Find the area of the entire skating rink.
(800:2 5=2000 m 2)
2) Winnie the Pooh collected x kg of honey from the hives, which is 30% of the amount he dreamed of. How much honey did Winnie the Pooh dream of? (x:30 100)
3) The boa constrictor gave the monkey “in” bananas, which is the amount he always gave. How much did he always give? (A)
Question for the class:
– What rule should we remember here?
(To find a number by its part expressed as a fraction, you can divide this part by the numerator and multiply by the denominator)3. Studying new material. “Discovery” of new knowledge by children.
(The didactic task is to organize and direct the cognitive activity of students towards the goal)
Today in the lesson we will try to find a simpler way to solve problems on finding a number from its fraction. The learned rules for multiplying and dividing fractions will help us with this.
– Write down the rule in your notebook (a = c: m n).
– Replace the division sign with a fraction line and try to write it down as one action with the number “a” and the fraction.
N = = in = in:
– Translate the resulting rule into mathematical language.
(To find a number by its part, you can divide this part by a fraction) Discovery. They repeated this rule to themselves.Now work in pairs:
Option 1 tells the rule to option 2, and option 2 to the first.– Why is this rule more convenient than the previous one? (The problem is solved with one action instead of
two)4. Physical education minute.
(The task is to relieve tension)
Find all the colors of the rainbow (every hunter wants to know where the pheasant is sitting). Colored squares are hung in different places around the classroom. To find the right color you need to spin around. Then exercise for the eyes.
Annex 1.
5. Primary consolidation.
(The didactic task is to get students to reproduce, comprehend, initially generalize and systematize new knowledge. Reinforce the methodology for the student’s upcoming answer during the next survey)
Primary consolidation takes place in the form of frontal work and work in pairs.
(with commentary in loud speech)1) Find the number if it is 10.
2) Find the number if 1% is 4.
In writing
(with commenting and writing on the board and in notebooks)1) Masha skied 500 m, which was the entire distance. What is the distance? (500:=800m)
2) The mass of dried fish is 55% of the mass of fresh fish. How much fresh fish do you need? To get 231 kg of jerky? (231:=420kg)
3) The mass of the strawberries in the first box is equal to the mass of the strawberries in the second box. How many kg of strawberries were in two boxes if the first box contained 24 kg of strawberries?
Work in pairs
(teamwork) Write an expression for the problems.1) On a beautiful summer morning, a kitten named Woof ate x sausages, which made up his daily diet. How many sausages does kitten Woof eat per day? (x:=sausages)
2) Dunno read 117 pages, which amounted to 9% of the magic book. How many pages are there in a magic book? (117:=1300str)
6. Initial check of understanding of what has been learned
(in the form of independent work with testing in class).(Didactic task– control of knowledge and elimination of gaps on this topic)
Call one person from each option, they will silently work on the wings of the board. Then we check the solution.
1 option
1) find the number if it is 21. (49)
2) find a number if 15% of it is x. ()
3) find the number if 0.88 equals 211.2. (240)
Option 2
1) find the number if it is 24. (64)
2) find a number if 20% of it is x. (5x)
3) find the number if 0.25 equals 6.25. (25)
Rate yourself: not a single mistake – “5”; 1 error – “4”; whoever has more mistakes should work on the mistakes.
7. Summing up the lesson.
(Didactic task– give an analysis and assessment of the success of achieving the goal and outline the prospects for further work). You made a discovery in class today
They came up with a new way to solve problems involving fractions, which means they succeeded seven times more than if I had told you everything myself (look again at the epigraph to our lesson)
Reflection.
(Didactic task - mobilization of students to reflect on their behavior, motivation, methods of activity, communication).Now guys, continue the sentence: Today in the lesson I learned... Today in the lesson I liked... Today in the lesson I repeated... Today in the lesson I consolidated... Today in the lesson I graded myself... What types of work caused difficulties and require repetition... In what knowledge I’m sure... Did the lesson help you advance in knowledge, skills, abilities in the subject... Who, on what, should still work on...
How effective was the lesson today... a smiling little man, if you liked the lesson and everything worked out, and a sad little man, if something else didn’t work out (on everyone’s desk there are pictures with little men).
6
. Homework(Comment, it is differentiated) (Didactic task - ensuring an understanding of the purpose, content and methods of completing homework).
Page 104-105. clause 18. No. 680; No. 683; No. 783(a, b)
Additional task No. 656. (for strong students).
For the creative group - come up with tasks on a new topic.
7. Grades for the lesson.
Everyone worked well and absorbed knowledge with gusto. Children! Thank you for the lesson.
“Methodology for teaching solving problems on finding fractions
from a number and a number by its fraction"
Most applications of mathematics involve the measurement of quantities. However, it is not always possible to perform division on a set of integers: a unit of a quantity does not always fit an integer number of times in the quantity being measured. In order to accurately express the measurement result in such a situation, it is necessary to expand the set of integers by introducing fractional numbers. People came to this conclusion in ancient times: the need to measure lengths, areas, masses and other quantities led to the emergence of fractional numbers.
Students are introduced to fractional numbers in the primary grades. The concept of a fraction is then refined and expanded upon in middle school. And one of the most difficult topics in high school math is solving fraction problems. Fractions have been taught at school for more than one year; there are several stages in studying the topic. This is due to various restrictions on the use of numbers. Therefore, the fifth grade program is closely intertwined with the sixth grade program. Problems that develop ideas about fractions are quite complex for students to understand, so when solving problems involving fractions, a mathematics teacher has to act outside the box, relying not only on traditional explanations.
Methods of teaching solving problems on finding a fraction from a number and a number from its fraction.
In the fifth grade, students have already learned to solve problems on finding a part of a number and on finding a number from its fraction. To solve these problems they applied the following rules:
1) To find the part of a number expressed as a fraction, you need to divide this number by the denominator and multiply by the numerator;
2) To find a number by its part expressed as a fraction, you need to divide this part by the denominator and multiply by the numerator.
In sixth grade, students learn that part of a number is found by multiplying by a fraction, and a number by its part is found by dividing by a fraction. Therefore, the teacher has the opportunity to eliminate gaps in students’ knowledge on this topic using material to consolidate new ways of solving problems on finding a part of a number and a number by its part.
When solving fraction problems, the main difficulty for students is determining the type of problem. The explanatory text of textbooks often does not contain a brief record of the conditions of these problems, and this leads students to misunderstand why in one case they must multiply a number by a fraction, and in another, divide a number by a given fraction. Therefore, when solving problems on finding a fraction from a number and a number from its fraction, it is necessary that students see what in the problem statement is a whole and what is its part.
1.Tasks on finding a fraction of a number.
Task 1.
20 trees should be planted on the school site. On the first day, the students planted. How many trees did they plant on the first day?
20 trees are 1 (whole).
This is that part of the trees (part of the whole),
which was planted on the first day.
20: 4 = 5, and all trees are equal
5 · 3 = 15, that is, 15 trees were planted on the site on the first day.
Answer: 15 trees were planted on the school site on the first day.
We write the solution to the problem using the expression: 20: 4 3 = 15.
20 was divided by the denominator of the fraction and the result was multiplied by the numerator.
The same result will be obtained if 20 is multiplied by .
(20 3) : 4 = 20 .
Conclusion: To find a fraction of a number, you need to multiply the number by the given fraction.
Task 2.
In two days, 20 km were paved. On the first day, 0.75 of this distance was paved. How many kilometers of the road were paved on the first day?
20 km is 1 (integer).
0.75 - this is that part of the road (part of the whole),
which was paved on the first day
Since 0.6 = then to solve the problem you need to multiply 20 by .
We get 20== =15. This means that on the first day 15 kilometers were paved.
You get the same answer if you multiply 20 by 0.75.
We have: 200.75=15.
Since percentages can be written as a fraction, problems of finding percentages of a number can be solved in a similar way.
Task 3.
In two days, 20 km were paved. On the first day, 75% of this distance was paved. How many kilometers of the road were paved on the first day?
20 km is 100%
Let us depict the entire plot of land in the form of a rectangle ABCD. The figure shows that the area occupied by apple trees occupies a plot of land. You can get the same answer if you multiply by:
Answer: The entire plot of land is occupied by apple trees.
Material for consolidating new ways of solving problems on finding a fraction from a number is best distributed into sections, in the first of which tasks on the direct implementation of the new rule are performed, then problems on finding a fraction from a number are analyzed, after which students move on to solving combined problems, the solution stage which is the solution to a simple fraction problem.
a) https://pandia.ru/text/80/420/images/image017_16.gif" width="19" height="49 src="> from 245; c) from 104; d) from https:// pandia.ru/text/80/420/images/image017_16.gif" width="19" height="49 src=">; m) 65% of 2.
1. 120 kg of potatoes were brought to the school canteen. On the first day, we used up all the potatoes we had brought. How many kilograms of potatoes did you use on the first day?
2. The length of the rectangle is 56 cm. The width is equal to the length. Find the width of the rectangle.
3. The school site covers an area of 600 m2. Sixth grade students dug up 0.3 of the entire site on the first day. How much area did the students dig on the first day?
4. There are 25 people in the drama club. Girls make up 60% of all club participants. How many girls are in the club?
5. Vegetable garden area hectares. The vegetable garden is planted with potatoes. How many hectares are planted with potatoes?
1. 2 kg of millet was poured into one bag, and this amount into the other.
How much less millet was poured into the second bag than into the first?
2. 2.7 tons of carrots were collected from one plot, and this amount from another. How many vegetables were collected from the two plots?
3. The bakery bakes 450 kg of bread per day. 40% of all bread goes to the retail chain, the rest goes to canteens. How many kg of bread goes to canteens every day?
4. 320 tons of vegetables were brought to the vegetable storehouse. 75% of the vegetables brought were potatoes, and the rest was cabbage. How many tons of cabbage were brought to the vegetable store?
5. The depth of the mountain lake at the beginning of summer was 60m. In June, its level decreased by 15%, and in July it became shallow by 12% from the June level. What was the depth of the lake by the beginning of August?
6. Before lunch, the traveler walked 0.75 of the intended path, and after lunch he walked the distance traveled before lunch. Did the traveler cover the entire intended route in one day?
7. 39 days were spent on repairing tractors in winter, and 7 days less on repairing combines. The repair time for trailed equipment was the same as the time it took to repair combine harvesters. How many days longer did the repair of tractors take than the repair of trailed equipment?
8. In the first week, the team completed 30% of the monthly norm, in the second - 0.8 of what was completed in the first week, and in the third week - of what was completed in the second week. What percentage of the monthly quota remains for the team to complete in the fourth week?
2. Finding a number by its fraction.
Problems of finding a number from its fraction are the inverse of problems of finding the fraction of a given number. If in problems of finding a fraction of a number a number was given and it was required to find some fraction of this number, then in these problems a fraction of a number was given and it was required to find this number itself.
Let us turn to solving problems of this type.
Task 1.
On the first day, the traveler walked 15 km, which was 5/8 of the entire journey. How far did the traveler have to travel?
Let's write down a short condition:
The entire distance is 1 (integer).
– this is 15 km
15 km is 5 shares. How many kilometers are in one lobe?
Since the entire distance contains 8 such parts, we find it:
3 8 = 24 (km).
Answer: The traveler must walk 24 km.
Let's write the solution to the problem by the expression: 15: 5 · 8 = 24(km) or 15: 5 · 8 = · 8 = = 15= 15:.
Conclusion: To find a number from a given value of its fraction, you need to divide this value by the fraction.
Task 2.
The captain of the basketball team accounts for 0.25 of all points scored in the game. How many total points did this team get in the game if the captain brought the team 24 points?
The entire number of points received by a team is 1 (integer).
45% is 9 squared notebooks
Since 45% = 0.45, and 9: 0.45 = 20, then we bought 20 notebooks in total.
It is also advisable to distribute material for consolidation in order to consolidate new ways of solving problems of finding a number by its fraction into sections. In the first section, tasks are completed to consolidate the new rule, in the second, problems of finding a number by its fraction are analyzed, and in the third, students analyze the solution of more complex problems, part of which are problems of finding a number by its fraction.
6) After replacing the engine, the average speed of the plane increased by 18%? Which is 68.4 km/h. What was the average speed of the plane with the same engine?
1) The length of the rectangle is https://pandia.ru/text/80/420/images/image005_25.gif" width="37" height="73"> of the entire cherry, in the second 0.4, and in the third - the rest 20 kg How many kilograms of cherries were collected?
5) Three workers produced a certain number of parts. The first worker produced 0.3 of all parts, the second - 0.6 of the remainder, and the third - the remaining 84 parts. How many parts did the workers make in total?
6) On the experimental plot, cabbage occupied the plot, potatoes occupied the remaining area, and the remaining 42 hectares were sown with corn. Find the area of the entire experimental plot.
7) The car covered the entire journey in the first hour, the remaining distance in the second hour, and the rest of the distance in the third hour. It is known that in the third hour he walked 40 km less than in the second hour. How many kilometers did the car travel in these three hours?
Fraction problems are an important tool for teaching mathematics. With their help, students gain experience working with fractional and integer quantities, comprehend the relationships between them, and gain experience in applying mathematics to solving practical problems. Solving fraction problems develops ingenuity and intelligence, the ability to pose and answer questions, and prepares schoolchildren for further education.
mathematic teacher
MBOU Lyceum No. 1 Nakhabino
Literature:
3. Didactic materials in mathematics: 5th grade: workshop/, . – M.: Akademkniga / Textbook, 2012.
4. Didactic materials in mathematics: 6th grade: workshop/, . – M.: Akademkniga/Textbook, 2012.
5. Independent and test work in mathematics for 6th grade. / , . – M.: ILEKSA, 2011.
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.
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Slide captions:
“Consider unhappy that day or that hour in which you did not learn anything new and did not add anything to your education” Y.A. Kamensky
Finding a number from a given value of its fraction Mathematics teacher Tokareva I.A. MBOU gymnasium No. 1 Lipetsk
Read the fractions: What is another name for them? Arrange these fractions in ascending order.
Find from 40; 2. How many decimeters are in half a meter? 3. Find the part of the smallest six-digit number. 4. How many hours are there in parts of the day?
5. How many seconds are there in parts of a minute? 6. How many minutes are there in a quarter of an hour? 7. There are 30 students in the class, some of them are good. How many good students are there in the class? 8. How many months does it contain?
9. The length of the wire is 64 m. Parts were cut off from it. How many meters of wire did you cut? (64 40 m) 10. We thought of a number that is equal to 15. What number did we think of? (15:3 5=25.)
Finding a number from a given value of its fraction Read the text of the textbook yourself, page 91, up to the example. Solve problem 10 in a new way. 10. We thought of a number that is equal to 15. What number did we think of?
Find the number if: What conclusion can you draw? (If the fraction is proper, then the number is greater than the value of the fraction; if the fraction is improper, then the number is less than the value of the fraction.)
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