Finding a number by a given value of its fraction. Finding a number by its fraction - Knowledge Hypermarket. Problems of finding the fraction of a number
Solving problems from the book of problems Vilenkin, Zhokhov, Chesnokov, Schwarzburd for grade 6 in mathematics on the topic:
§ 3. Multiplication and division of ordinary fractions:
18. Finding a number by its fraction
1 2/5 of the skating rink was cleared of snow, which is 800 m2. Find the area of the entire ice rink.
SOLUTION
2 Wheat has been sown on 2,400 hectares. which is 0.8 of the total field. Find its area.
SOLUTION
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?
SOLUTION
647 The girl skied 300 m, which was 3/8 of the entire distance. How long is the distance?
SOLUTION
648 The pile rises 1.5 m above the water, which is 3/16 of the length of the entire pile. What is its length
SOLUTION
649 211.2 tons of grain were sent to the elevator, which is 0.88 grain, threshed per day. How much grain was threshed in a day?
SOLUTION
650 After replacing the engine, the average speed of the aircraft increased by 18%, which is 68.4 km / h. What was the average speed of the aircraft with the previous engine.
SOLUTION
651 The mass of dried fish is 55% of the mass of fresh fish. How much fresh do you need to take to get 231 kg of jerky?
SOLUTION
652 The mass of grapes in the first box is 7/9 of the mass of grapes in the second. How many kilograms of grapes were in two boxes if there were 21 kg of grapes in the first?
SOLUTION
653 3/8 of the skis received by the store were sold, leaving 120 pairs of skis. How many pairs did the store receive?
SOLUTION
654 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?
SOLUTION
655 The bank bought several shares of the plant and a year later sold them for 576.8 million rubles, receiving 3% of the profit. How much did the bank spend on the purchase of shares?
SOLUTION
656 On the first day, tourists covered 5/24 of the planned route, and on the second - 0.8 of what they covered on the first day. How long is the planned path if on the second day the tourists walked 24 km?
SOLUTION
657 The student read 75 pages first, and then a few more pages. Their number was 40% of the first read. How many pages are in the book if 3/4 of the book has been read in total?
SOLUTION
658 The cyclist first covered 12 1/4 km, and then a few more kilometers, which was 3/7 of the first leg of the route. After that, he had to drive 2/3 of the entire way. What is its length
SOLUTION
659 3/5 of 12 is 1/4 of the unknown number. Find this number.
SOLUTION
660 35% of 128.1 is 49% of the unknown number. Find him
SOLUTION
661 The kiosk sold 40% of all notebooks on the first day, 53% on the second, and the remaining 847 on the third. How many notebooks did the kiosk sell in three days?
SOLUTION
662 The vegetable base on the first day released 40% of all available potatoes, on the second 60% of the remainder, and on the third the remaining 72 tons. How many tons of potatoes were there at the base?
SOLUTION
663 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?
SOLUTION
664 On the first day, the tractor team plowed 3/8 of the plot, on the second 2/5 of the remainder, and on the third the remaining 216 hectares. Determine the area of the site.
SOLUTION
665 The car covered 4/9 of the entire journey in the first hour, 3/5 of the remaining journey in the second hour, and the rest of the journey in the third. It is known that in the third hour it covered 40 km less than in the second. How many kilometers did the car cover in these 3 hours?
SOLUTION
666 Perform calculations. Find with the help of a microcalculator the number, 12.7% of which is equal to 4.5212; number, 8.52% of which is 3.0246.
SOLUTION
668 Without performing division, compare.
SOLUTION
669 How many times less than its inverse number: 1/5; 2/3; 1/6; 0.3?
SOLUTION
670 Think of a number that is 4 times less than your inverse; 9 times.
SOLUTION
671 Verbally divide the center number by the numbers in circles.
SOLUTION
672 How many square tiles with a side of 20 cm are 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.
SOLUTION
673 Find the rule for placing numbers in semicircles and insert the missing numbers
SOLUTION
675 The cyclist covered 7 1/2 km in 3/5 hours. How many kilometers will a cyclist travel in 2 1/2 hours if he travels at the same speed
SOLUTION
676 In 1/3 hour, a pedestrian covered 1 1/2 km. How many kilometers will a pedestrian cover in 2 1/2 hours if he walks at the same speed?
SOLUTION
678 Find the meaning of the expression
SOLUTION
679 Perform steps 10.1 + 9.9 107.1: 3.5: 6.8 - 4.85; 12.3 + 7.7 * 187.2: 4.5: 6.4 - 3.4
SOLUTION
680 7/12 of the kerosene contained there was poured out of the barrel. How many liters of kerosene was in the barrel if 84 liters were poured out of it
SOLUTION
681 Volodya read 234 pages, which is 36% of the entire book. How many pages are in this book?
SOLUTION
682 Using a new tractor to plow the field resulted in a 70% time saving and took 42 hours. How long would it take to get the job done on an old tractor?
SOLUTION
683 A pillar dug into the ground 2/13 of its length rises 5 1/2 m above the ground. Find the length of the pillar.
SOLUTION
684 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?
SOLUTION
685 Point C divides segment AB into two segments AC and CB. The length AC is 0.65 times the length of the segment CB. Find CB and AB if AC = 3.9 cm.
SOLUTION
686 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 second is 5/12 of the length of the first section. What is the length of the entire distance if the length of the second section is 5 km? How long is the third one?
SOLUTION
687 From a full barrel they took 14.4 kg of sauerkraut and then another 5/12 of this amount. After that, 5/8 of the sauerkraut that was there earlier remained in the barrel. How many kilograms of cabbage were in a full barrel?
SOLUTION
688 When Kostya had covered 0.3 all the way from home to school, he still had 150 m to go to the middle of the way. How long was the path from home to school?
SOLUTION
689 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 - the remaining 104. How many trees were planted in total?
SOLUTION
690 There were turning, milling and grinding machines in the shop. Lathes accounted for 5/11 of all these machines. The number of grinders is 2/5 of the number of lathes. How many machines of these types were there in the workshop, if there are 8 less milling machines than turning ones?
SOLUTION
691 Perform actions (1.704: 0.8 - 1.73) · 7.16 - 2.64; 227.36: (865.6 - 20.8 * 40.5) * 8.38 + 1.12; (0.9464: (3.5 * 0.13) + 3.92) * 0.18; 275.4: (22.74 + 9.66) * (937.7 - 30.6 * 30.5).
Finding a number by its fraction
Remark 1
To find a number for a given value of its fraction, you need to divide this value by a fraction.
Example 1
Anton earned in a week of study three quarters excellent marks. How many marks did Anton get if there were excellent marks? 6 .
Solution.
By the problem statement, $ 6 $ marks are $ \ frac (3) (4) $.
Let's find the number of all marks:
$ 6 \ div \ frac (3) (4) = 6 \ cdot \ frac (4) (3) = \ frac (6 \ cdot 4) (3) = \ frac (2 \ cdot 3 \ cdot 4) (3) = 2 \ cdot 4 = $ 8.
Answer: only $ 8 $ marks.
Example 2
Mowed $ \ frac (4) (9) $ wheat in the field. Find the area of the field, if it was cut $ 36 $ ha.
Solution.
By the hypothesis of the problem, $ 36 $ ga is $ \ frac (4) (9) $.
Find the area of the entire field:
$ 36 \ div \ frac (4) (9) = 36 \ cdot \ frac (9) (4) = \ frac (36 \ cdot 9) (4) = \ frac (4 \ cdot 9 \ cdot 9) (4) = $ 81.
Answer: the area of the entire field is $ 81 $ ha.
Example 3
In one day, the bus passed the $ \ frac (2) (3) $ route. Find the duration of the planned route if the bus traveled $ 350 km per day?
Solution.
By the problem statement, $ 350 $ km is $ \ frac (2) (3) $.
Let's find the duration of the entire bus route:
$ 350 \ div \ frac (2) (3) = 350 \ cdot \ frac (3) (2) = \ frac (350 \ cdot 3) (2) = 175 \ cdot 3 = 525 $.
Answer: duration of the planned route $ 525 $ km.
Example 4
The worker raised his labor productivity by $% \ $ and made $ 24 more parts in the same period than planned. Find the number of parts scheduled to be completed by the worker.
Solution.
By the condition of the problem, $ 24 $ parts = $ 8 \% $, and $ 8 \% = $ 0.08.
Let's find the number of parts planned for execution by the worker:
$ 24 \ div 0.08 = 24 \ div \ frac (8) (100) = 24 \ cdot \ frac (100) (8) = \ frac (24 \ cdot 100) (8) = \ frac (3 \ cdot 8 \ cdot 100) (8) = 300 $.
Answer: planned $ 300 $ parts for the worker.
Example 5
In the workshop, $ 9 $ machines were repaired, which is $ 18 \% $ of all machines in the workshop. How many machines are there in the workshop?
Solution.
By the condition of the problem, $ 9 $ machines = $ 18 \% $, and $ 18 \% = 0.18. $
Let's find the number of machines in the workshop:
$ 9 \ div 0.18 = 9 \ div \ frac (18) (100) = 9 \ cdot \ frac (100) (18) = \ frac (9 \ cdot 100) (18) = \ frac (9 \ cdot 100 ) (2 \ cdot 9) = \ frac (100) (2) = 50 $.
Answer: in the workshop $ 50 $ machines.
Fractional expressions
Consider the fraction $ \ frac (a) (b) $, which is equal to the quotient $ a \ div b $. In this case, it is convenient to write the quotient from dividing one expression by another using a line.
Example 6
For example, the expression $ (13.5–8.1) \ div (20.2 + 29.8) $ can be written as follows:
$ \ frac (13.5-8.1) (20.2 + 29.8) $.
After performing the calculations, we get the value of this expression:
$ \ frac (13.5-8.1) (20.2 + 29.8) = \ frac (5.4) (50) = \ frac (10.8) (100) = 0.108 $.
Definition 1
Fractional expression is called the quotient of two numbers or numerical expressions in which the $ ":" $ sign is replaced by a fractional bar.
Example 7
$ \ frac (2,4) (1,3 \ cdot 7,5) $, $ \ frac (\ frac (5) (8) + \ frac (3) (11)) (2.7-1.5 ) $, $ \ frac (2a-3b) (3a + 2b) $, $ \ frac (5,7) (ab) $ are fractional expressions.
Definition 2
A numeric expression that is written above the slash is called numerator, and the numerical expression, which is written below the fractional bar, is denominator fractional expression.
The numerator and denominator of a fractional expression can contain numbers, numeric or literal expressions.
For fractional expressions, the same rules apply as for ordinary fractions.
Example 8
Find the value of the expression $ \ frac (5 \ frac (3) (11)) (3 \ frac (2) (7)) $.
Solution.
Multiply the numerator and denominator of this fractional expression by $ 77 $:
$ \ frac (5 \ frac (3) (11)) (3 \ frac (2) (7)) = \ frac (5 \ frac (3) (11) \ cdot 77) (3 \ frac (2) ( 7) \ cdot 77) = \ frac (406) (253) = 1.6047 ... $
Answer: $ \ frac (5 \ frac (3) (11)) (3 \ frac (2) (7)) = 1.6047… $
Example 9
Find the product of two fractional numbers $ \ frac (16,4) (1,4) $ and $ 1 \ frac (3) (4) $.
Solution.
$ \ frac (16,4) (1,4) \ cdot 1 \ frac (3) (4) = \ frac (16,4) (1,4) \ cdot \ frac (7) (4) = \ frac (4.1) (0.2) = \ frac (41) (2) = $ 20.5.
Answer: $ \ frac (16.4) (1.4) \ cdot 1 \ frac (3) (4) = 20.5 $.
"Teaching methodology for solving problems to find a fraction
from the number and the number according to its fraction "
Most of the applications of mathematics are related to the measurement of quantities. However, it is not always possible to perform division on a set of integers: it is not always possible for a unit of a quantity to fit an integer number of times in a measured quantity. In order to accurately express the measurement result in such a situation, it is necessary to expand the set of integers by entering 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.
Acquaintance of students with fractional numbers occurs in elementary grades. Then the concept of a fraction is refined and expanded in secondary school. And one of the most difficult topics in high school mathematics is solving problems on fractions. Fractions take place at school for more than one year, in the study of the topic there are several stages. This is due to various restrictions on the use of numbers. Therefore, the fifth grade curriculum is closely intertwined with the sixth grade curriculum. The tasks on which the idea of fractions is formed are quite difficult for students to perceive, therefore, when solving problems with fractions, the mathematics teacher has to act outside the box, relying not only on traditional explanations.
A method of teaching the solution of problems to find a fraction of a number and a number by its fraction.
In the fifth grade, students have already learned how to solve problems to find a part of a number and to find a number by 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 fractional part, you need to divide this part by the denominator and multiply by the numerator.
In the sixth grade, students learn that part of a number is found by multiplying by a fraction, and the number by its part is by dividing by a fraction. Therefore, the teacher has the opportunity to eliminate gaps in the knowledge of students on this topic on the material to consolidate new ways of solving problems to find a part of a number and a number in its part.
When solving problems on fractions, the main difficulties for students are caused by the definition of the type of problems. In the explanatory text of textbooks, there is often no short record of the conditions of these problems, and this leads students to a misunderstanding of why, in one case, they must multiply a number by a fraction, and in the other, divide a number by a given fraction. Therefore, when solving problems on finding a fraction of a number and a number by its fraction, it is necessary that the students see what in the problem statement is whole and what is part of it.
1. Problems of finding a fraction of a number.
Objective 1.
20 trees are to be planted on the school grounds. On the first day, the disciples were seated. How many trees did they plant on the first day?
20 trees is 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 by the expression: 20: 4 3 = 15.
Divide 20 by the denominator of the fraction and multiply the result by the numerator.
You get the same result if you multiply 20 by.
(20 3): 4 = 20.
Output: to find a fraction of a number, you need to multiply the number by the given fraction.
Objective 2.
In two days we have asphalted 20 km. On the first day, 0.75 of this distance was asphalted. How many kilometers of 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 asphalted 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 15 kilometers were asphalted on the first day.
You get the same answer if you multiply 20 by 0.75.
We have: 200.75 = 15.
Since the percentage can be written as a fraction, the problem of finding the percentage of the number is solved in a similar way.
Objective 3.
In two days we have asphalted 20 km. On the first day, 75% of this distance was asphalted. How many kilometers of road were paved on the first day?
20 km is 100%
Let's represent the entire land plot in the form of a rectangle ABCD. It can be seen from the figure that the area occupied by apple trees occupies a land plot. The same answer can be obtained by multiplying by:
Answer: the entire land plot is occupied by apple trees.
The material for consolidating new ways of solving problems on finding a fraction of a number is best distributed into sections, in the first of which tasks are performed on the direct implementation of a new rule, then tasks for finding a fraction of a number are analyzed, after which students move on to solving combined problems, the stage of solving which is the solution to a simple problem on fractions.
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 cafeteria. On the first day, we used up all the potatoes we 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 length. Find the width of the rectangle.
3. The school area covers an area of 600 m2. Pupils of the sixth grade on the first day dug up 0.3 of the entire site. What area did the students dig up on the first day?
4. There are 25 people in the drama circle. Girls make up 60% of all circle members. How many girls are in the club?
5. The area of the garden is ha. Vegetable gardens are planted with potatoes. How many hectares are planted with potatoes?
1. In one bag poured 2 kg of millet, and in the other - this amount.
How much less millet was poured into the second bag than into the first one?
2. 2.7 tons of carrots were harvested from one site, and this amount was collected from the other. How many vegetables were collected from the two sites?
3. The bakery bakes 450 kg of bread per day. 40% of all bread goes to the distribution network, the rest goes to canteens. How many kg of bread go to the canteens every day?
4. 320 tons of vegetables were brought to the vegetable store. 75% of the imported vegetables were potatoes, and the remainder was cabbage. How many tons of cabbage were brought to the vegetable store?
5. The depth of the mountain lake by the beginning of summer was 60m. In June, its level dropped by 15%, and in July it became shallower 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 planned path, and after lunch he walked the path that had been traversed before lunch. Did the traveler walk the entire planned path in a day?
7. It took 39 days to repair tractors in winter, and 7 days less to repair combines. The repair time for trailed implements was the same as for the repair of the combines. How many days longer did tractor repairs take than repairing trailed implements?
8. In the first week, the brigade completed 30% of the monthly norm, in the second - 0.8 of what was done in the first week, and in the third week - what was done in the second week. How many percent of the monthly norm is left for the team to complete in the fourth week?
2. Finding a number by its fraction.
The problems of finding a number by its fraction are inverse to the problems of finding the fraction of a given number. If in the problems of finding the fraction of a number a number was given and it was required to find a certain fraction of this number, then in these problems a fraction of the number is given and it is required to find this number itself.
Let us turn to solving problems of this type.
Objective 1.
On the first day, the traveler covered 15 km, which was 5/8 of the entire journey. How far did the traveler have to travel?
Let's write a short condition:
All distance is 1 (integer).
- this is 15 km
15 km is 5 stakes. How many kilometers are in one share?
Since the entire distance contains 8 such parts, we will find it:
3 8 = 24 (km).
Answer: the traveler must walk 24 km.
Let's write down the solution of the problem by the expression: 15: 5 8 = 24 (km) or 15: 5 8 = 8 = = 15 = 15 :.
Output: to find a number for a given value of its fraction, this value must be divided by a fraction.
Objective 2.
The captain of the basketball team accounts for 0.25 of all points earned in the game. How many points did this team score in the game if the captain earned the team 24 points?
The total number of points received by the team is 1 (integer).
45% is 9 squared notebooks
Since 45% = 0.45, and 9: 0.45 = 20, then a total of 20 notebooks were bought.
It is also advisable to distribute the material for consolidation to consolidate new ways of solving problems of finding a number by its fraction. In the first section, tasks are carried out to consolidate the new rule, in the second, the tasks of finding a number by its fraction are dealt with, and in the third, students analyze the solution of more complex problems, part of which are the tasks of finding a number by its fraction.
6) After replacing the engine, did the average speed of the aircraft increase by 18%? Which is 68.4 km / h. What was the average speed of the aircraft with the previous engine?
1) The length of the rectangle is https://pandia.ru/text/80/420/images/image005_25.gif "width =" 37 "height =" 73 "> all cherries, in the second 0.4, and in the third - the rest 20 kg How many kilograms of cherries were harvested in total?
5) Three workers made a number of parts. The first worker made 0.3 of all the parts, the second made 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 of the remaining area, and the remaining 42 hectares were sown with corn. Find the area of the entire test site.
7) The car passed in the first hour of the whole journey, in the second hour - the remaining way, and in the third hour - the rest of the way. It is known that in the third hour he walked 40 km less than in the second hour. How many kilometers did the car go in those three hours?
Fraction problems are an important math teaching tool. With their help, students gain experience in working with fractional and whole quantities, comprehend the relationship between them, and gain experience in applying mathematics to solving practical problems. Solving problems on fractions develops ingenuity and ingenuity, the ability to pose questions, answer them, and prepares schoolchildren for further education.
mathematic teacher
MBOU Lyceum No. 1 n. Nakhabino
Literature:
3. Didactic materials in mathematics: Grade 5: workshop /,. - M .: Akademkniga / Textbook, 2012.
4. Didactic materials in mathematics: 6th grade: workshop /,. - M .: Akademkniga / Textbook, 2012.
5. Independent and test works in mathematics for the 6th grade. /,. - M .: ILEKSA, 2011.
Class: 6
Lesson presentations
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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:
- Organizing time.
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).
15; ; 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; v:; :; :; (; ; ; NS)
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? (a)
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 in a notebook (a = b: m n).
- 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.
Annex 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 mass of strawberries in the first box is equal to the mass of 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 x is 20% of it. (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
came 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 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.
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 us again have a 5% discount card. 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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