The rule for finding a number by its fraction:

To find a number for a given value of its fraction, you need to divide this value by a fraction.

Let's consider how to find a number by its fraction, with specific examples.

Examples.

1) Find the number 3/4 of which are 12.

To find a number by its fraction, divide this number by this fraction. To, you need to multiply the given number by the inverse of the fraction (that is, by the inverted fraction). To, you need to multiply the numerator by this number, and leave the denominator unchanged. 12 and 3 by 3. Since the denominator is one, the answer is an integer.

2) Find a number if 9/10 is 3/5.

To find a number for a given value of its fraction, divide this value by this fraction. To divide a fraction into 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. Reduce 10 and 5 by 5, 3 and 9 - by 3. As a result, we got the correct irreducible fraction, which means this is the final result.

3) Find a number whose 9/7 are equal

To find a number based on the value of its fraction, divide this value by this fraction. Mixed number and multiply it by the inverse of the second (inverted fraction). Reduce 99 and 9 by 9, 7 and 14 - by 7. Since we received an incorrect fraction, it is necessary to select the whole part from it.

In this lesson, we will look at the types of tasks for shares and percentages. We will learn how to solve these problems and find out which of them we can face in real life. Let's find out the general algorithm for solving similar problems.

We do not know what the number was initially, but we know how much it turned out when a certain fraction was taken from it. You need to find the starting point.

That is, we do not know, but we also know.

Example 4

The grandfather spent his life in the village, which was 63 years. How old is grandfather?

We do not know the original number - age. But we know the proportion and how many years this proportion is from the age. We make up 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 the grandfather will give out such information about his years of life.

But the following situation is very common.

Example 5

Discount in the store with the card 5%. 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 percentage that is written on the card) and how much the discount was.

We compose our standard line. We express the unknown quantity and find it.

Answer: 600 rubles.

Example 6

We are faced with such a task even more often. We see not the amount of the discount, but what the cost is after applying the discount. And the question is the same: how much would we pay without a discount?

Let's say we have a 5% discount card again. We showed the card at the checkout and paid 1140 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 initial 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. Grade 6 / N. Ya. Vilenkin and V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd. - M .: Mnemosina, 2011. Pp. 104-105. Clause 18. No. 680; No. 683; No. 783 (a, b)

2. Mathematics. Grade 6 / N. Ya. Vilenkin and V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd. - M .: Mnemosina, 2011. No. 656.

3. The program of sports school competitions included long jump, high jump and running. All participants took part in the running competition, 30% of all participants in the long jump, and the remaining 34 students in the high jump competition. Find the number of competitors.

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Slide captions:

“Consider the day or the hour unhappy when you have not learned anything new and added nothing to your education” Ya.A. Kamensky

Finding a number by a given value of its fraction Mathematics teacher Tokareva I.A. MBOU gymnasium number 1 in Lipetsk

Read the fractions: How else can you call them? Arrange these fractions in ascending order.

Find from 40; 2. How many decimeters are in half a meter? 3. Find the fraction of the smallest six-digit number. 4. How many hours are there in parts of a day?

5. How many seconds are in parts of a minute? 6. How many minutes are in a quarter of an hour? 7. There are 30 students in the class, some of them are good. How many good guys are there in the class? 8. How many months does it contain

9. The length of the wire is 64 m. Parts were cut from it. How many meters of wire have you cut? (64 40 m) 10. Have a number that is equal to 15. What number have you in mind? (15: 3 5 = 25.)

Finding a number by a given value of its fraction Read the text of the textbook p. 91 yourself before an example. Solve problem 10 in a new way. 10. Have conceived a number, which is 15. What number have you conceived?

Find the number if: What conclusion can be drawn? (If the fraction is correct, then the number is greater than the value of the fraction; if the fraction is incorrect, then the number is less than the value of the fraction.)

On the subject: methodological developments, presentations and notes

Mathematics lesson in grade 6 Topic Division of fractions. Solving problems of finding a number by a given value of its fraction.

Mathematics lesson in grade 6 Topic Division of fractions. Solving problems of finding a number by a given value ...

Finding a number by its fraction. Finding a fraction of a number.

Presentation for the lesson. To generalize and systematize knowledge on the topics of finding a number by its fraction and finding a fraction of a number ...

Presentation for the mathematics lesson "Finding a number by a given value of its fraction"

The presentation contains the goals and objectives of the lesson, examples of tasks for finding a number by a given value of its fraction ...

Class: 6

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Epigraph to the lesson:

“The one who studies independently, succeeds seven times more than the one to whom everything is explained” (Arthur Giterman, German poet)

Lesson type: lesson in learning new material.

Methods: partial search.

Forms: individual, collective, group, individual.

(A place - 1 lesson on the topic)

Lesson type: explanatory and illustrative

The purpose of the lesson: to come up with a new way of solving problems in fractions, to consolidate the skills and abilities of solving problems.

• to systematize the solution of problems into parts, to deduce a new method of solving problems for finding a number according to its part.

• to help the development of students' interest not only in the content, but also in the process of mastering knowledge, to 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 the lesson. Create a positive motivation for schoolchildren to perform mental and practical actions. Education of responsibility, organization, perseverance in solving tasks.

Equipment: illustrative material, presentation for the lesson. Sheets with an assignment for reflection, a textbook on mathematics Mathematics. Grade 6 / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S.I.Shvartsburd. Moscow: Mnemosina, 2011.

Lesson plan:

1. Organizing time.

Basic knowledge actualization and their correction.

Learning new knowledge.

Physical education.

Primary anchoring.

Primary test of understanding of what has been learned.

Summing up the lesson. Reflection.

Homework.

Estimates.

During the classes

1. Organizational moment.

(Didactic task - psychological attitude of students)

Hello, sit down. We communicate the topic, the purpose of the lesson and the practical meaning of the topic.

The purpose of our lesson is to come up with a new way to solve problems with fractions.

2. Actualization of basic knowledge and their correction

(The didactic task is to prepare students for work in the classroom. Providing motivation and acceptance by students of the goal, educational and cognitive activities, updating basic knowledge and skills).

fifteen; ; 3 6; ; (2;; 19; c)

Questions to 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 distribution property of multiplication or convert a mixed number to an improper fraction)

- How to multiply mixed numbers?

2): 2; in:; :; :; (;;; x)

Questions to 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 desks for the weak group:

Repeat the algorithms for solving problems to find a number by its part.

1) Cleared of snow from the skating rink, which is 800 m 2. Find the area of ​​the entire ice rink.

(800: 2 5 = 2000 m 2)

2) Winnie collected x kg of honey from the hives, which is 30% of the amount he dreamed of. How much honey have you dreamed of, Winnie the Pooh? (x: 30 100)

3) The boa constrictor gave the monkey "v" bananas, which is the amount that he always gave. How much did he always give? (but)

Question to the class:

- What rule should be remembered here?

(To find a number by its fractional part, you can divide this part by the numerator and multiply by the denominator)

3. Learning 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 an easier way to solve problems of finding a number by its fraction. The learned rules for multiplying and dividing fractions will help us with this.

- Write down the rule (a = b: m n) in a notebook.

- Replace the division sign with a slash and try to write it down as one action with the number "a" and a 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) Opening. We repeated this rule to ourselves.

Now work in pairs:

Option 1 tells the rule to option 2, and option 2 tells the first option.

- Why is this rule more convenient than the previous one? (The problem is solved with one action instead of

two)

4. Physical education.

(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 posted in different places in the classroom. To find the right color, you need to twist. Then charge for the eyes.

Attachment 1.

5. Primary anchoring.

(The didactic task is to achieve from students the reproduction, awareness, primary generalization and systematization of new knowledge. Consolidation of the methodology of the student's forthcoming answer during the next survey)

Primary reinforcement 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 commentary and writing on the board and in notebooks)

1) Masha skied 500 m, which was the entire distance. How long is the distance? (500: = 800m)

2) The mass of dried fish is 55% of the mass of fresh fish. How much fresh fish to take. To get 231 kg of jerky? (231: = 420kg)

3) The weight of the strawberries in the first box is the same as the weight of the strawberries in the second box. How many kg of strawberries were there in two boxes if there were 24 kg of strawberries in the first box?

Working in pairs

(teamwork) Make up an expression for the tasks.

1) On a beautiful summer morning, a kitten named Woof ate x sausages, which made up his daily diet. How many sausages does a Woof kitten eat in a day? (x: = sausages)

2) Dunno read 117 pages, which was 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 verification in the classroom).

(Didactic task- knowledge control and elimination of gaps on this topic)

One person from each call option, they will silently work on the wings of the board. Then we check the solution.

Option 1

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 is 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 is 6.25. (25)

Assess yourself: not a single mistake - “5”; 1 error - “4”; who has more mistakes - make work on the mistakes.

7. Summing up the lesson.

(Didactic task- to analyze and evaluate the success of achieving the goal and outline the prospect of further work). You made a discovery in class today

have come up with a new way of solving problems on fractions, which means they have succeeded seven times more than if I had told you everything myself (we look again at the epigraph to our lesson)

Reflection.

(Didactic task - mobilizing students to reflect on their behavior, motivation, methods of activity, communication).

And now the guys continue the sentence: Today in the lesson I learned ... Today in the lesson I liked it ... Today in the lesson I repeated ... Today in the lesson I reinforced ... Today in the lesson I gave myself an assessment ... What types of work caused difficulties and require repetition ... In what knowledge I'm sure ... Did the lesson help me to advance in knowledge, skills, skills in the subject ... Who, on, what else should be worked on ...

How effective was the lesson today ... smiling little man, if you liked the lesson and everything worked out and a sad little man, if still, something doesn't work out (everyone has pictures with little people on their desks).

6

. Homework

(Comment, it is differentiated) (Didactic task - ensuring an understanding of the purpose, content and ways of doing homework).

P. 104-105. Clause 18. # 680; # 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, absorbed knowledge with appetite. Children! Thank you for the lesson.

The whole skating rink.

Decision. Let's denote the area of ​​the ice rink through x m 2. By the condition, this area is equal to 800 m 2, that is, x = 800.
Hence, x = 800: = 800 = 2000. The area of ​​the ice rink is 2000 m 2.

To find a number for a given value of its fraction, you need to divide this value by a fraction.

Objective 2. Wheat has been sown on 2,400 hectares, which is 0.8 of the entire field. Find the area of ​​the entire field.

Decision. Since 2400: 0.8 = 24000: 8 = 3000, the area of ​​the entire field is 3000 hectares.

Objective 3. Having increased labor productivity by 7%, the worker made 98 more parts during the same period than planned. How many parts did the worker have to make according to the plan?

Decision. Since 7% = 0.07, and 98: 0.07 = 1400, the worker, according to the plan, had to make 1400 parts.

? Formulate a rule for finding a number for a given value of it fractions... Tell me how to find a number based on a given percentage.

TO 631. The girl skied 300 m, which was the entire distance. How long is the distance?

632. The pile rises 1.5 m above the water, which is the length of the entire pile. How long is the entire pile?

633. 211.2 tons of grain were sent to the elevator, which is 0.88 grain, threshed per day. How much grain was threshed in a day?

634. For the rationalization proposal, the engineer received 68.4 rubles in excess of the monthly salary, which is 18% of this salary. What is the monthly salary of an engineer?

635. The mass of dried fish is 55% of the mass of fresh fish. How much fresh fish do you need to take to get 231 kg of dried fish?

636. The mass of grapes in the first box is equal to the mass of grapes in the second box. How many kilograms of grapes were in two boxes if there were 21 kilograms of grapes in the first box?

637. The skis received by the store were sold, after which 120 pairs of skis remained. How many pairs of skis did the store receive?

638. When dried, potatoes lose 85.7% of their weight. How many raw potatoes do you need to take to get 71.5 tons of dried?

639. A Sberbank depositor made a certain amount on a fixed-term deposit, and a year later he had 576 rubles in his savings account. 80 k. What was the amount of the deposit if Sberbank pays 3% per annum on time deposits?

640. On the first day, the tourists walked the planned path, and on the second day 0.8 of what they passed on the first day. How long is the planned path if on the second day the tourists walked 24 km?

641. The student read 75 pages first, and then a few more pages. Their number was 40% of the first read. How many pages are there in the book if the whole book is read?

642. The cyclist first traveled 12 km, and then a few more kilometers, which was from the first leg of the route. After that, he had to drive all the way. How long is the entire path?

643. of the number 12 is an unknown number. Find this number.

644. 35% of 128D is 49% of unknown number. Find this number.

645. On the first day, the kiosk sold 40% of all notebooks, on the second day 53% of all notebooks, and on the third day - the remaining 847 notebooks. How many notebooks did the kiosk sell in three days?

646. The vegetable base on the first day released 40% of all available potatoes, on the second day 60% of the remainder, and on the third day - the remaining 72 tons. How many tons of potatoes were there at the base?

647. Three workers made a number of parts. The first worker made 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?

648. On the first day the tractor brigade plowed the plot, on the second day the remainder, and on the third day the remaining 216 hectares. Determine the area of ​​the site.
649. The car passed in the first hour of the entire journey, in the second hour of the remaining journey, and in the third hour the rest of the journey. It is known that in the third hour it covered 40 km less than in the second hour. How many kilometers did the car cover in these 3 hours?

650. It is possible to find a number by a given value of its percent using a microcalculator. For example, to find a number whose 2.4% is 7.68, you can use the following program :Perform calculations. Find with a microcalculator:
a) a number, 12.7% of which is equal to 4.5212;
b) a number, 8.52% of which is equal to 3.0246.

P 651. Calculate orally:

652. Without performing division, compare:

653. How many times less than its reciprocal:

654. Think of a number that is 4 times less than your inverse; 9 times.

655. Divide orally the central number by the number in circles:

656. How many square tiles with a side of 20 cm will be needed to lay the floor in a room that is 5.6 m long and 4.4 m wide. Solve the problem in two ways.

M 657. Find the rule for placing numbers in semicircles and insert the missing numbers (fig. 29).

658. Perform division:

659. The cyclist covered 7 km in an hour. How many kilometers will a cyclist travel in 2 hours if he travels at the same speed?

660. In 4 hours a pedestrian walked 1 km. How many kilometers will a pedestrian cover in 2 hours if he walks at the same speed?

661. Reduce the fraction:

663. Perform actions:

1) 10,14-9,9 107,1:3,5:6,8-4,8;
2) 12,34-7,7 187,2:4,5:6,4-3,4.

D 664. Kerosene was poured from the barrel. How many liters of kerosene was in the barrel if 84 liters were poured out of it?

665. When buying a color TV on credit, 234 rubles were paid in cash, which is 36% of the cost of the TV. How much does a TV cost?

666. The worker received a ticket to a sanatorium with a 70% discount and paid 42 rubles for it. How much is a ticket to the sanatorium?

667. The pillar, dug into the ground at its length, rises 5 m above the ground. Find the entire length of the pillar.

668. The turner, turning 145 parts on the lathe, exceeded the plan by 16%. How many parts did you have to cut according to the plan?

669. Point C divides segment AB into two segments AC and CB. The length of the segment AC is 0.65 of the length of the segment CB. Find the lengths of the segments CB and AB if AC = 3.9 cm.

670. The skiing distance is divided into three sections. The length of the first section is 0.48 of the length of the entire distance, the length of the second section is the length of the Lervo section. What is the length of the entire distance if the length of the second section is 5 km? How long is the third section?

671. From a full barrel they took 14.4 kg of sauerkraut and then this amount. After that, the sauerkraut that was there earlier remained in the barrel. How many kilograms of sauerkraut were in a full barrel?

672. When Kostya passed 0.3 all the way from home to school, he still had 150 m to go to the middle of the way. How long is the way from Kostya's house to school?

673. Three groups of schoolchildren have planted trees along the road. The first group planted 35% of all available trees, the second - 60% of the remaining trees, and the third group - the remaining 104 trees. How many trees have been planted in total?

674. The shop had turning, milling and grinding machines. Lathes made up all of these machines. The number of grinding machines was the number of lathes. How many machines of these types were in the workshop if there were 8 less milling machines than lathes?

675. Proceed as follows:

a) (1.704: 0.8 -1.73) 7.16 -2.64;
b) 227.36: (865.6 - 20.8 40.5) 8.38 + 1.12;
c) (0.9464: (3.5 0.13) + 3.92) 0.18;
d) 275.4: (22.74 + 9.66) (937.7 - 30.6 30.5).

N.Ya. Vilenkin, A.S. Chesnokov, S.I. Schwarzburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school

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