Roundabout tasks. A cyclist left point A of the circular track (cm). How to solve? From point a of the circular road 30 92
Sections: Maths
Lesson type: repetitive and generalizing lesson.
Lesson objectives:
- educational - repeat methods for solving various types of word problems for movement
- developing - develop students 'speech through the enrichment and complication of its vocabulary, develop students' thinking through the ability to analyze, generalize and systematize material
- educational - the formation of a humane attitude among students towards participants in the educational process
Lesson equipment:
- interactive board;
- envelopes with tasks, thematic control cards, cards - consultants.
Lesson structure.
The main stages of the lesson |
Tasks to be solved at this stage |
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| Organizing time, introductory part |
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| Preparing students for active work (review) |
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| The stage of generalization and systematization of the studied material (work in groups) |
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| Checking the performance of the work, adjusting (if necessary) |
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| Summing up the lesson. Parsing homework |
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Forms of organizing the cognitive activity of students:
- frontal form of cognitive activity - at stages II, IY, Y.
- group form of cognitive activity - at stage III.
Teaching methods: verbal, visual, practical, explanatory - illustrative, reproductive, partly - search, analytical, comparative, generalizing, traductive.
During the classes
I. Organizational moment, introductory part.
The teacher announces the topic of the lesson, the objectives of the lesson, and the highlights of the lesson. Checks that the class is ready for work.
II. Preparing students for active work (review)
Answer the questions.
- What kind of movement is called uniform (movement with constant speed).
- What is the formula for the path with uniform motion ( S \u003d Vt).
- Express speed and time from this formula.
- Specify units of measurement.
- Speed \u200b\u200bUnit Conversion
III. The stage of generalization and systematization of the studied material (work in groups)
The whole class is divided into groups (5-6 people per group). It is desirable that there are students in one group different levels preparation. Among them, a group leader (the strongest student) is appointed, who will lead the work of the group.
All groups receive envelopes with assignments (they are the same for all groups), consultant cards (for weak students) and thematic control sheets. On the sheets of thematic control, the group leader gives marks to each student of the group for each task and notes the difficulties that the students encountered in completing specific tasks.
Card with tasks for each group.
№ 5.
No. 7. The motor boat passed 112 km against the river and returned to the point of departure, spending 6 hours less on the way back. Find the current speed if the boat speed in still water is 11 km / h. Give your answer in km / h.
No. 8. The motor ship goes along the river to its destination 513 km and after anchorage returns to the point of departure. Find the speed of the motor ship in still water, if the current speed is 4 km / h, the stop lasts 8 hours, and the ship returns to the point of departure 54 hours after leaving it. Give your answer in km / h.
Sample of a thematic control card.
| Class ________ Name of student ___________________________________ | ||
Job No. |
Comment |
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Consultants cards.
| Card number 1 (consultant) |
| 1. Driving on a straight road |
| When solving problems of uniform motion, two situations often occur. If the initial distance between the objects is S, and the velocities of the objects are V1 and V2, then: a) when objects move towards each other, the time after which they will meet is equal. b) when objects move in one direction, the time after which the first object will catch up with the second is equal to, ( V 2 > V 1) |
Example 1. The train, having covered 450 km, was stopped due to a snow drift. Half an hour later, the path was cleared, and the driver, increasing the speed of the train by 15 km / h, brought it to the station without delay. Find the initial speed of the train if the distance it traveled to the stop was 75% of the total distance.
X \u003d -75 does not fit the problem, where x\u003e 0. Answer: the initial speed of the train is 60 km / h. |
| Card number 2 (consultant) |
2. Driving on a closed road |
| If the length of the closed road is S, and the speed of objects V 1 and V 2, then: a) when objects move in different directions, the time between their encounters is calculated by the formula; |
| Example 2.In competitions on the ring track, one skier completes a lap 2 minutes faster than the other, and after an hour he went around him exactly one lap. How long does it take for each skier to complete the lap? Let be Sm - the length of the ring track and xm / min and ym / min - speeds of the first and second skiers, respectively ( x\u003e y) . Then S / xmin and S / ymin is the time it takes for the first and second skiers to complete the lap, respectively. From the first condition we obtain the equation. Since the speed of removal of the first skier from the second skier is ( x - y) m / min, then from the second condition we have the equation. Let's solve the system of equations.
Let's make a replacement S / x \u003d aand S / y \u003d b, then the system of equations will take the form:
60(a + 2) – 60a \u003d a(a + 2)a 2 + 2a -120 \u003d 0. The quadratic equation has one positive root a \u003d10 then b \u003d12. This means that the first skier completes the lap in 10 minutes, and the second skier in 12 minutes. Answer: 10 minutes; 12 minutes |
| Card number 3 (consultant) |
3. Driving along the river |
| If an object moves along the river, then its speed is equal to Vflux. \u003d Vsob. +
V leak. If the object is moving against the flow of the river, then its speed is V against flow \u003d V sob. - Vflow The intrinsic speed of the object (speed in still water) is The river speed is The speed of the raft is equal to the speed of the river. |
| Example 3.The boat went down the river for 50 km, and then went in the opposite direction for 36 km, which took him 30 minutes longer than downstream. What is own speed boats, if the river speed is 4 km / h? Let the boat's own speed be xkm / h, then its speed along the river is ( x + 4) km / h, and against the river flow ( x - 4) km / h. The time of movement of the boat along the river is equal to hours, and against the river flow is hours. Since 30 minutes \u003d 1/2 hour, then according to the condition of the problem, we will compose the equation \u003d. Multiply both sides of the equation by 2 ( x + 4)(x- 4) >0 . We get 72 ( x + 4) -100(x - 4) = (x + 4)(x - 4) x 2 + 28x- 704 \u003d 0 x 1 \u003d 16, x 2 \u003d - 44 (exclude, since x\u003e 0). So, the boat's own speed is 16 km / h. Answer: 16 km / h. |
IV. The stage of parsing the solution of problems.
Tasks that have caused difficulties for students are analyzed.
No. 1. From two cities, the distance between which is 480 km, two cars drove towards each other at the same time. How many hours will the cars meet if their speeds are 75 km / h and 85 km / h?
- 75 + 85 \u003d 160 (km / h) - approach speed.
- 480: 160 \u003d 3 (h).
Answer: the cars will meet in 3 hours.
No. 2. From cities A and B, the distance between which is 330 km, two cars drove towards each other at the same time and met 3 hours later at a distance of 180 km from the city B. Find the speed of the car that left the city A. Give your answer in km / h.
- (330 - 180): 3 \u003d 50 (km / h)
Answer: the speed of a car leaving city A is 50 km / h.
No. 3. From point A to point B, the distance between which is 50 km, a motorist and a cyclist left at the same time. It is known that a motorist drives 65 km more per hour than a cyclist. Determine the speed of the cyclist if it is known that he arrived at point B 4 hours 20 minutes later than the motorist. Give your answer in km / h.
Let's make a table.
Let's make an equation, considering that 4 hours 20 minutes \u003d
,Obviously, x \u003d -75 does not fit the problem.
Answer: the speed of the cyclist is 10 km / h.
No. 4. Two motorcyclists start simultaneously in the same direction from two diametrically opposite points of the circular track, the length of which is 14 km. In how many minutes will motorcyclists level up for the first time if one of them is 21 km / h faster than the other?
Let's make a table.
Let's make an equation.
where 1/3 hour \u003d 20 minutes.Answer: In 20 minutes the motorcyclists will level up for the first time.
No. 5. From one point of the circular track, the length of which is 12 km, two cars started simultaneously in the same direction. The speed of the first car is 101 km / h, and 20 minutes after the start, it was ahead of the second car by one lap. Find the speed of the second car. Give your answer in km / h.
Let's make a table.
Let's make an equation.
Answer: the speed of the second car is 65 km / h.
No. 6. A cyclist left point A of the circular track, and 40 minutes later a motorcyclist followed him. Eight minutes after the departure, he caught up with the cyclist for the first time, and 36 minutes after that he caught up with him a second time. Find the speed of the biker if the track is 30 km long. Give your answer in km / h.
Let's make a table.
Movement to the first meeting |
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cyclist |
No. 9. From pier A to pier B, the distance between which is 168 km, the first motor ship departed at a constant speed, and 2 hours after that, the second one followed it at a speed 2 km / h higher. Find the speed of the first ship if both ships arrived at point B at the same time. Give your answer in km / h. Let's draw up a table based on their condition that the speed of the first motor ship is x km / h. Let's make the equation: Multiplying both sides of the equation by x ,Answer: the speed of the first motor ship is equal to the river 12 km / h V. Summing up the lesson.While summing up the results of the lesson, the students' attention should once again be drawn to the principles of solving movement problems. When giving homework, give an explanation of the most difficult tasks. Literature. 1) Article : Mathematics of the Unified State Exam 2014 (a system of problems from an open bank of tasks) A.G. Koryanov, N.V. Nadezhkina - published on the website | ||
Problem 1. From point A to point B two cars drove out simultaneously.
The first drove all the way at a constant speed.
The second drove the first half of the way at a speed,
lower speed of the first by 14 km / h,
and the second half of the journey at a speed of 105 km / h,
and therefore arrived at B at the same time as the first car.
Find the speed of the first car,
if it is known that it is more than 50 km / h.
Solution: Let's take the whole distance as 1.
Let's take the speed of the first car as x.
Then, the time during which the first car drove the entire distance
equally 1 / x.
The second vehicle speed for the first half of the journey, i.e. 1/2,
was 14 km / h less than the speed of the first car, x-14.
The time it took for the second car is 1/2: (x-14) \u003d 1/2 (x-14).
The second half of the journey, i.e. 1/2, car passed
at a speed of 105 km / h.
The time it took is 1/2: 105 \u003d 1/2 * 105 \u003d 1/210.
The times of the first and second are equal to each other.
Let's make the equation:
1 / x \u003d 1/2 (x-14) + 1/210
Find the common denominator - 210x (x-14)
210 (x-14) \u003d 105x + x (x-14)
210x - 2940 \u003d 105x + x² - 14x
x² - 119x + 2940 \u003d 0
Solving this quadratic equation through the discriminant, we find the roots:
x1 \u003d 84
х2 \u003d 35. The second root does not fit according to the problem statement.
Answer: the speed of the first car is 84 km / h.
Problem 2. From point A of the circular route, the length of which is 30 km,
two motorists started at the same time in one direction.
The speed of the first is 92 km / h, and the speed of the second is 77 km / h.
How many minutes will the first motorist take
will be ahead of the second by 1 circle?
Decision: This task, despite the fact that it is given in grade 11,
can be solved at the level primary school.
We'll ask four questions in total and get four answers.
1. How many kilometers will the first motorist cover in 1 hour?
92 km.
2. How many kilometers will the second motorist cover in 1 hour?
77 km.
3. How many kilometers will the first motorist lead the second after 1 hour?
92 - 77 \u003d 15 km.
4. How many hours will it take for the first motorist to be 30 km ahead of the second?
30:15 \u003d 2 hours \u003d 120 minutes.
Answer: in 120 minutes.
Problem 3. From point A to point B, the distance between which is 60 km,
a motorist and a cyclist left at the same time.
It is known that at one o'clock a motorist passes
90 km more than a cyclist.
Determine the speed of the cyclist if it is known that he arrived at point B 5 hours 24 minutes later than the motorist.
Solution: To correctly solve any task assigned to us,
you must adhere to a certain plan.
And the most important thing is to understand what we want from this.
That is, what equation do we want to arrive at under the given conditions.
We will compare each other's time.
A car travels 90 km per hour more than a cyclist.
This means the speed of the car is greater than the speed
a cyclist at 90 km / h.
Taking the cyclist's speed as x km / h,
we get the speed of the car x + 90 km / h.
Cyclist travel time 60 / h.
Travel time by car - 60 / (x + 90).
5 hours 24 minutes is 5 24/60 hours \u003d 5 2/5 \u003d 27/5 hours
Let's make the equation:
60 / x \u003d 60 / (x + 90) + 27/5 Reduce the numerator of each fraction by 3
20 / x \u003d 20 / (x + 90) + 9/5 Common denominator 5x (x + 90)
20 * 5 (x + 90) \u003d 20 * 5x + 9x (x + 90)
100x + 9000 \u003d 100x + 9x² + 810x
9x² + 810x - 9000 \u003d 0
x² + 90x - 1000 \u003d 0
Solving this equations through the discriminant or Vieta's theorem, we get:
х1 \u003d - 100 Does not fit the meaning of the problem.
x2 \u003d 10
Answer: the speed of the cyclist is 10 km / h.
Problem 4. The cyclist drove 40 km from city to village.
On the way back he rode at the same speed
but after 2 hours drive made a stop for 20 minutes.
After stopping, he increased the speed by 4 km / h
and therefore spent as much time on the way back from village to city as on the way from city to village.
Find the initial speed of the cyclist.
Solution: we solve this problem in relation to the time spent
first to the village and then back.
The cyclist traveled from town to village at the same speed x km / h.
In doing so, he spent 40 / x hours.
He drove 2 km back in 2 hours.
He has 40 - 2 km left to drive, which he covered
at a speed of x + 4 km / h.
At the same time, the time that he spent on the way back
consists of three terms.
2 hours; 20 minutes \u003d 1/3 hour; (40 - 2x) / (x + 4) hours.
Let's make the equation:
40 / x \u003d 2 + 1/3 + (40 - 2x) / (x + 4)
40 / x \u003d 7/3 + (40 - 2x) / (x + 4) Common denominator 3x (x + 4)
40 * 3 (x + 4) \u003d 7x (x + 4) + 3x (40 - 2x)
120x + 480 \u003d 7x² + 28x + 120x - 6x²
x² + 28x - 480 \u003d 0 Solving this equations through the discriminant or Vieta's theorem, we get:
x1 \u003d 12
х2 \u003d - 40 Does not match the problem statement.
Answer: The initial speed of the cyclist is 12 km / h.
Problem 5. Two cars left the same point at the same time in the same direction.
The speed of the first is 50 km / h, the second is 40 km / h.
Half an hour later, a third car left the same point in the same direction,
who overtook the first car 1.5 hours later,
than the second car.
Find the speed of the third car.
Solution: In half an hour, the first car will travel 25 km, and the second 20 km.
Those. the initial distance between the first and third car is 25 km,
and between the second and third - 20 km.
In the event that one car catches up with another, their speeds are deducted.
If we take the speed of the third car as x km / h,
then it turns out that he caught up with the second car in 20 / (x-40) hours.
Then he will catch up with the first car in 25 / (x - 50) hours.
Let's make the equation:
25 / (x - 50) \u003d 20 / (x - 40) + 3/2 Common denominator 2 (x - 50) (x - 40)
25 * 2 (x - 40) \u003d 20 * 2 (x - 50) + 3 (x - 50) (x - 40)
50x - 2000 \u003d 40x - 2000 + 3x² - 270x + 6000
3x² - 280x + 6000 \u003d 0 Solving this equation through the discriminant, we get
x1 \u003d 60
x2 \u003d 100/3
Answer: the speed of the third car is 60 km / h.
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« A bicycle left point A of the circular track.»- found 251 tasks
(impressions: 606 , answers: 13 )
A cyclist left point A of the circular track, and 10 minutes later a motorcyclist followed him. 2 minutes after departure, he caught up with the cyclist for the first time, and 3 minutes after that he caught up with him a second time. Find the speed of the biker if the track is 5 km long. Give your answer in km / h.
Quest B14 ()(impressions: 625 , answers: 11 )
A cyclist left point A of the circular track, and 20 minutes later a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 10 minutes later he caught up with him a second time. Find the speed of the biker if the track is 10 km long. Give your answer in km / h.
The correct answer has not yet been determined
Quest B14 ()(impressions: 691 , answers: 11 )
A cyclist left point A of the circular track, and 10 minutes later a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 15 minutes after that he caught up with him a second time. Find the speed of the biker if the track is 10 km long. Give your answer in km / h.
Answer: 60
Quest B14 ()(impressions: 613 , answers: 11 )
A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 47 minutes after that he caught up with him a second time. Find the speed of the biker if the track is 47 km. Give your answer in km / h.
The correct answer has not yet been determined
Quest B14 ()(impressions: 610 , answers: 9 )
A cyclist left point A of the circular track, and 20 minutes later a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 19 minutes after that he caught up with him a second time. Find the speed of the biker if the track is 19 km long. Give your answer in km / h.
The correct answer has not yet been determined
Quest B14 ()(impressions: 618 , answers: 9 )
A cyclist left point A of the circular track, and 20 minutes later a motorcyclist followed him. 2 minutes after the departure, he caught up with the cyclist for the first time, and 30 minutes later he caught up with him a second time. Find the speed of the biker if the track is 50 km. Give your answer in km / h.
The correct answer has not yet been determined
Quest B14 ()(impressions: 613 , answers: 9 )
A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 5 minutes after the departure, he caught up with the cyclist for the first time, and 26 minutes later he caught up with him a second time. Find the speed of the biker if the track is 39 km. Give your answer in km / h.
The correct answer has not yet been determined
Quest B14 ()(impressions: 622 , answers: 9 )
A cyclist left point A of the circular track, and 50 minutes later a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 12 minutes later he caught up with him a second time. Find the speed of the motorcyclist if the track is 20 km long. Give your answer in km / h.
The correct answer has not yet been determined
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This work A cyclist left point A of the circular track, and 30 minutes later a motorcyclist followed him. After 10 minutes (Control) on the subject (Macroeconomics and Public Administration), it was custom-made by the specialists of our company and passed its successful defense. Job - A cyclist left point A of the circular track and a motorcyclist followed 30 minutes later. After 10 minutes on the subject Macroeconomics and Public Administration, it reflects its topic and the logical component of its disclosure, the essence of the issue under study is revealed, the main provisions and leading ideas of this topic are highlighted.
Job - A cyclist left point A of the circular track and a motorcyclist followed 30 minutes later. After 10 minutes, it contains: tables, drawings, the latest literary sources, the year of delivery and protection of the work - 2017. In work A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. After 10 minutes (Macroeconomics and public administration) the relevance of the research topic is revealed, the degree of development of the problem is reflected, on the basis of a deep assessment and analysis of scientific and methodological literature, in the work on the subject Macroeconomics and public administration, the object of analysis and its issues are comprehensively considered, both from theoretical, and the practical side, the goal and specific tasks of the topic under consideration are formulated, there is a logic of presentation of the material and its sequence.
