Grade 5 Introduction to Probability (4 hours)
(development of 4 lessons on this topic)
learning goals : - introduce the definition of a random, reliable and impossible event;
Lead the first ideas about solving combinatorial problems: using a tree of options and using the multiplication rule.
educational goal: development of students' mindset.
Development goal : development of spatial imagination, improvement of the skill of working with a ruler.
Reliable, Impossible and Random Events (2 hours)
Combinatorial tasks (2 hours)
Reliable, impossible and random events.
First lesson
Lesson equipment: dice, coin, backgammon.
Our life is largely made up of accidents. There is such a science "Probability Theory". Using its language, it is possible to describe many phenomena and situations.
Even the primitive leader understood that a dozen hunters had a greater “probability” of hitting a bison with a spear than one. Therefore, they hunted collectively then.
Such ancient commanders as Alexander the Great or Dmitry Donskoy, preparing for battle, relied not only on the valor and skill of warriors, but also on chance.
Many people love mathematics for the eternal truths twice two is always four, the sum of even numbers is even, the area of a rectangle is equal to the product of its adjacent sides, etc. In any problem that you solve, everyone gets the same answer - you just need to make no mistakes in the solution.
Real life is not so simple and unambiguous. The outcomes of many events cannot be predicted in advance. It is impossible, for example, to say for sure which side a coin tossed will fall, when the first snow will fall next year, or how many people in the city will want to make a phone call within the next hour. Such unpredictable events are called random .
However, the case also has its own laws, which begin to manifest themselves with repeated repetition of random phenomena. If you toss a coin 1000 times, then the "eagle" will fall out about half the time, which cannot be said about two or even ten tosses. "Approximately" does not mean half. This, as a rule, may or may not be the case. The law generally does not state anything for sure, but gives a certain degree of certainty that some random event will occur. Such regularities are studied by a special branch of mathematics - Probability theory . With its help, you can predict with a greater degree of confidence (but still not sure) both the date of the first snowfall and the number of phone calls.
Probability theory is inextricably linked with our daily life. This gives us a wonderful opportunity to establish many probabilistic laws empirically, repeatedly repeating random experiments. The materials for these experiments will most often be an ordinary coin, a dice, a set of dominoes, backgammon, roulette, or even a deck of cards. Each of these items is related to games in one way or another. The fact is that the case here appears in the most frequent form. And the first probabilistic tasks were associated with assessing the chances of players to win.
Modern probability theory has moved away from gambling, but their props are still the simplest and most reliable source of chance. By practicing with a roulette wheel and a die, you will learn how to calculate the probability of random events in real life situations, which will allow you to assess your chances of success, test hypotheses, and make optimal decisions not only in games and lotteries.
When solving probabilistic problems, be very careful, try to justify each step, because no other area of mathematics contains such a number of paradoxes. Like probability theory. And perhaps the main explanation for this is its connection with the real world in which we live.
In many games, a die is used, which has a different number of points from 1 to 6 on each side. The player rolls the die, looks at how many points have fallen (on the side that is located on top), and makes the appropriate number of moves: 1,2,3 ,4,5, or 6. Throwing a die can be considered an experience, an experiment, a test, and the result obtained can be considered an event. People are usually very interested in guessing the onset of an event, predicting its outcome. What predictions can they make when a dice is rolled? First prediction: one of the numbers 1,2,3,4,5, or 6 will fall out. Do you think the predicted event will come or not? Of course it will definitely come. An event that is sure to occur in a given experience is called reliable event.
Second prediction : the number 7 will fall out. Do you think the predicted event will come or not? Of course it won't, it's just impossible. An event that cannot occur in a given experiment is called impossible event.
Third Prediction : the number 1 will fall out. Do you think the predicted event will come or not? We are not able to answer this question with complete certainty, since the predicted event may or may not occur. An event that may or may not occur in a given experience is called random event.
Exercise : describe the events that are discussed in the tasks below. As certain, impossible or random.
We toss a coin. The coat of arms appeared. (random)
The hunter shot at the wolf and hit. (random)
The student goes for a walk every evening. During a walk, on Monday, he met three acquaintances. (random)
Let's mentally carry out the following experiment: turn a glass of water upside down. If this experiment is carried out not in space, but at home or in a classroom, then water will pour out. (authentic)
Three shots fired at the target. There were five hits" (impossible)
We throw the stone up. The stone remains suspended in the air. (impossible)
The letters of the word "antagonism" are rearranged at random. Get the word "anachroism". (impossible)
№959. Petya conceived natural number. The event is as follows:
a) an even number is conceived; (random) b) an odd number is conceived; (random)
c) a number is conceived that is neither even nor odd; (impossible)
d) a number that is even or odd is conceived. (authentic)
№ 961. Petya and Tolya compare their birthdays. The event is as follows:
a) their birthdays do not match; (random) b) their birthdays are the same; (random)
d) both birthdays fall on holidays - New Year (January 1) and Independence Day of Russia (June 12). (random)
№ 962. When playing backgammon, two dice are used. The number of moves that a participant in the game makes is determined by adding the numbers on the two faces of the die that have fallen out, and if a “double” falls out (1 + 1.2 + 2.3 + 3.4 + 4.5 + 5.6 + 6), then the number of moves is doubled. You roll the dice and calculate how many moves you have to make. The event is as follows:
a) you must make one move; b) you must make 7 moves;
c) you must make 24 moves; d) you must make 13 moves.
a) - impossible (1 move can be made if the combination 1 + 0 falls out, but there is no number 0 on the dice).
b) - random (if 1 + 6 or 2 + 5 falls out).
c) - random (if the combination 6 +6 falls out).
d) - impossible (there are no combinations of numbers from 1 to 6, the sum of which is 13; this number cannot be obtained even when a “double” is rolled, because it is odd).
Check yourself. (math dictation)
1) Indicate which of the following events are impossible, which are certain, which are random:
Football match "Spartak" - "Dynamo" will end in a draw. (random)
You will win by participating in the win-win lottery (authentic)
Snow will fall at midnight, and the sun will shine 24 hours later. (impossible)
There will be a math test tomorrow. (random)
You will be elected President of the United States. (impossible)
You will be elected president of Russia. (random)
2) You bought a TV in a store, for which the manufacturer gives a two-year warranty. Which of the following events are impossible, which are random, which are certain:
The TV will not break within a year. (random)
The TV won't break for two years. (random)
Within two years, you won't have to pay for TV repairs. (authentic)
The TV will break in the third year. (random)
3) A bus carrying 15 passengers has 10 stops to make. Which of the following events are impossible, which are random, which are certain:
All passengers will get off the bus at different stops. (impossible)
All passengers will get off at the same stop. (random)
At every stop, someone will get off. (random)
There will be a stop at which no one will get off. (random)
At all stops, an even number of passengers will get off. (impossible)
At all stops, an odd number of passengers will get off. (impossible)
Homework : 53 No. 960, 963, 965 (come up with two reliable, random and impossible events yourself).
Second lesson.
Examination homework. (orally)
a) Explain what certain, random and impossible events are.
b) Indicate which of the following events is certain, which is impossible, which is random:
There will be no summer holidays. (impossible)
The sandwich will fall butter side down. (random)
The school year will eventually end. (authentic)
I will be asked in class tomorrow. (random)
I'm meeting a black cat today. (random)
№ 960. You opened this textbook to any page and chose the first noun that came across. The event is as follows:
a) there is a vowel in the spelling of the chosen word. ((authentic)
b) in the spelling of the chosen word there is a letter "o". (random)
c) there are no vowels in the spelling of the chosen word. (impossible)
d) the spelling of the chosen word has soft sign. (random)
№ 963. You are playing backgammon again. Describe the following event:
a) the player must make no more than two moves. (impossible - with a combination smallest numbers 1 + 1 player makes 4 moves; combination 1 + 2 gives 3 moves; all other combinations give more than 3 moves)
b) the player must make more than two moves. (reliable - any combination gives 3 or more moves)
c) the player must make no more than 24 moves. (reliable - the combination of the largest numbers 6 + 6 gives 24 moves, and all the rest - less than 24 moves)
d) the player must make a two-digit number of moves. (random - for example, a combination of 2 + 3 gives a one-digit number of moves: 5, and the fall of two fours gives a two-digit number of moves)
2. Problem solving.
№ 964. There are 10 balls in a bag: 3 blue, 3 white and 4 red. Describe the following event:
a) 4 balls are taken out of the bag, and all of them are blue; (impossible)
b) 4 balls are taken out of the bag, and they are all red; (random)
c) 4 balls were taken out of the bag, and they all turned out to be of different colors; (impossible)
d) 4 balls are taken out of the bag, and there is no black ball among them. (authentic)
Task 1 . The box contains 10 red, 1 green and 2 blue pens. Two items are taken at random from the box. Which of the following events are impossible, which are random, which are certain:
a) two red handles are taken out (random)
b) two green handles are taken out; (impossible)
c) two blue handles are taken out; (random)
d) handles of two different colors are taken out; (random)
e) two handles are taken out; (authentic)
e) Two pencils are taken out. (impossible)
Task 2. Winnie the Pooh, Piglet and everyone - everyone - everyone sits down at a round table to celebrate a birthday. With what number of all - all - all the event "Winnie the Pooh and Piglet will sit side by side" is reliable, and with what - random?
(if there are only 1 of all - all - all, then the event is reliable, if more than 1, then it is random).
Task 3. Out of 100 charity lottery tickets, 20 winning ones How many tickets do you need to buy to make the "you win nothing" event impossible?
Task 4. There are 10 boys and 20 girls in the class. Which of the following events are impossible for such a class, which are random, which are certain
There are two people in the class who were born in different months. (random)
There are two people in the class who were born in the same month. (authentic)
There are two boys in the class who were born in the same month. (random)
There are two girls in the class who were born in the same month. (authentic)
All boys were born in different months. (authentic)
All girls were born in different months. (random)
There is a boy and a girl born in the same month. (random)
There is a boy and a girl born in different months. (random)
Task 5. There are 3 red, 3 yellow, 3 green balls in a box. Draw 4 balls at random. Consider the event "Among the drawn balls there will be balls of exactly M colors". For each M from 1 to 4, determine which event it is - impossible, certain or random, and fill in the table:
Independent work.
Ioption
a) your friend's birthday is less than 32;
c) there will be a math test tomorrow;
d) Next year, the first snow in Moscow will fall on Sunday.
Throw a dice. Describe the event:
a) the cube, having fallen, will stand on its edge;
b) one of the numbers will fall out: 1, 2, 3, 4, 5, 6;
c) the number 6 will fall out;
d) a number that is a multiple of 7 will come up.
A box contains 3 red, 3 yellow and 3 green balls. Describe the event:
a) all drawn balls are of the same color;
b) all drawn balls of different colors;
c) among the drawn balls there are balls of different colors;
c) among the drawn balls there is a red, yellow and green ball.
IIoption
Describe the event in question as certain, impossible, or random:
a) a sandwich that has fallen off the table will fall on the floor, butter-side down;
b) snow will fall in Moscow at midnight, and in 24 hours the sun will shine;
c) you win by participating in a win-win lottery;
d) next year in May, the first spring thunder will be heard.
All two-digit numbers are written on the cards. One card is chosen at random. Describe the event:
a) the card turned out to be zero;
b) there is a number on the card that is a multiple of 5;
c) there is a number on the card that is a multiple of 100;
d) the card contains a number greater than 9 and less than 100.
The box contains 10 red, 1 green and 2 blue pens. Two items are taken at random from the box. Describe the event:
a) two blue handles are taken out;
b) two red handles are taken out;
c) two green handles are taken out;
d) green and black handles are taken out.
Homework: 1). Come up with two reliable, random and impossible events.
2). A task . There are 3 red, 3 yellow, 3 green balls in a box. We draw N balls at random. Consider the event "among the drawn balls there will be balls of exactly three colors." For each N from 1 to 9, determine which event it is - impossible, certain or random, and fill in the table:
combinatorial tasks.
First lesson
Checking homework. (orally)
a) We check the problems that the students came up with.
b) additional task.
I am reading an excerpt from V. Levshin's book "Three Days in Karlikanii".
“First, to the sounds of a smooth waltz, the numbers formed a group: 1+ 3 + 4 + 2 = 10. Then the young skaters began to change places, forming more and more new groups: 2 + 3 + 4 + 1 = 10
3 + 1 + 2 + 4 = 10
4 + 1 + 3 + 2 = 10
1 + 4 + 2 + 3 = 10 etc.
This continued until the skaters returned to their original position.
How many times have they changed places?
Today in the lesson we will learn how to solve such problems. They're called combinatorial.
3. Learning new material.
Task 1. How many two-digit numbers can be formed from the numbers 1, 2, 3?
Solution: 11, 12, 13
31, 32, 33. Only 9 numbers.
When solving this problem, we enumerated all possible options, or, as they usually say in these cases. All possible combinations. Therefore, such tasks are called combinatorial. It is quite common to calculate possible (or impossible) options in life, so it is useful to get acquainted with combinatorial problems.
№ 967. Several countries have decided to use for their national flag symbols in the form of three horizontal stripes of the same width in different colors - white, blue, red. How many countries can use such symbols, provided that each country has its own flag?
Solution. Let's assume that the first stripe is white. Then the second stripe can be blue or red, and the third stripe, respectively, red or blue. It turned out two options: white, blue, red or white, red, blue.
Let now the front page of blue color, then again we get two options: white, red, blue or blue, red, white.
Let the first stripe be red, then two more options: red, white, blue or red, blue, white.
There are 6 possible options in total. This flag can be used by 6 countries.
So, when solving this problem, we were looking for a way to enumerate possible options. In many cases, it turns out to be useful to construct a picture - a scheme for enumerating options. This is, first of all, illustrative Secondly, allows us to take into account everything, not to miss anything.
This scheme is also called a tree of possible options.
Front page
Second lane
third lane
Received combination
№ 968. How many two-digit numbers can be made from the numbers 1, 2, 4, 6, 8?
Solution. For two-digit numbers of interest to us, any of the given digits can be in the first place, except for 0. If we put the number 2 in the first place, then any of the given digits can be in the second place. There will be five two-digit numbers: 2.,22, 24, 26, 28. Similarly, there will be five two-digit numbers with the first digit 4, five two-digit numbers with the first digit 6 and five two-digit numbers with the first digit 8.
Answer: There are 20 numbers in total.
Let's build a tree of possible options for solving this problem.
Double figures
First digit
Second digit
Received numbers
20, 22, 24, 26, 28, 60, 62, 64, 66, 68,
40, 42, 44, 46, 48, 80, 82, 84, 86, 88.
Solve the following problems by constructing a tree of possible options.
№ 971. The leadership of a certain country decided to make its national flag like this: on a one-color rectangular background, a circle of a different color is placed in one of the corners. It was decided to choose colors from three possible ones: red, yellow, green. How many variants of this flag
exists? The figure shows some of the possible options.
Answer: 24 options.
№ 973. a) How many three-digit numbers can be made from the numbers 1,3, 5,? (27 numbers)
b) How many three-digit numbers can be made from the numbers 1,3, 5, provided that the numbers should not be repeated? (6 numbers)
№ 979. Modern pentathletes compete for two days in five sports: show jumping, fencing, swimming, shooting, and running.
a) How many options are there for the order of passing the types of competition? (120 options)
b) How many options are there for the order of passing the events of the competition, if it is known that the last event should be a run? (24 options)
c) How many options are there for the order of passing the types of competition, if it is known that the last type should be running, and the first - show jumping? (6 options)
№ 981. Two urns contain five balls in each five various colors: white, blue, red, yellow, green. One ball is drawn from each urn at a time.
a) how many different combinations of drawn balls are there (combinations like "white-red" and "red-white" are considered the same)?
(15 combinations)
b) How many combinations are there in which the drawn balls are of the same color?
(5 combinations)
c) how many combinations are there in which the drawn balls are of different colors?
(15 - 5 = 10 combinations)
Homework: 54, No. 969, 972, come up with a combinatorial problem ourselves.
№ 969. Several countries have decided to use symbols in the form of three vertical stripes of the same width in different colors for their national flag: green, black, yellow. How many countries can use such symbols, provided that each country has its own flag?
№ 972. a) How many two-digit numbers can be formed from the numbers 1, 3, 5, 7, 9?
b) How many two-digit numbers can be made from the numbers 1, 3, 5, 7, 9, provided that the numbers should not be repeated?
Second lesson
Checking homework. a) No. 969 and No. 972a) and No. 972b) - build a tree of possible options on the board.
b) verbally check the compiled tasks.
Problem solving.
So, before that, we have learned how to solve combinatorial problems using a tree of options. Is this a good way? Probably yes, but very cumbersome. Let's try to solve home problem No. 972 in a different way. Who can guess how this can be done?
Answer: For each of the five colors of T-shirts, there are 4 colors of shorts. Total: 4 * 5 = 20 options.
№ 980. The urns contain five balls each in five different colors: white, blue, red, yellow, green. One ball is drawn from each urn at a time. Describe the following event as certain, random, or impossible:
a) drawn balls of different colors; (random)
b) drawn balls of the same color; (random)
c) black and white balls are drawn; (impossible)
d) two balls are taken out, and both are colored in one of the following colors: white, blue, red, yellow, green. (authentic)
№ 982. A group of tourists plans to make a trip along the route Antonovo - Borisovo - Vlasovo - Gribovo. From Antonovo to Borisovo you can raft down the river or walk. From Borisovo to Vlasovo you can walk or ride bicycles. From Vlasovo to Gribovo you can swim along the river, ride bicycles or walk. How many hiking options can tourists choose? How many hiking options can tourists choose, provided that at least one of the sections of the route they must use bicycles?
(12 route options, 8 of them using bicycles)
Independent work.
1 option
a) How many three-digit numbers can be made from the numbers: 0, 1, 3, 5, 7?
b) How many three-digit numbers can be made from the numbers: 0, 1, 3, 5, 7, provided that the numbers should not be repeated?
Athos, Porthos and Aramis have only a sword, a dagger and a pistol.
a) In how many ways can the musketeers be armed?
b) How many weapon options are there if Aramis must wield a sword?
c) How many weapon options are there if Aramis should have a sword and Porthos should have a pistol?
Somewhere, God sent a piece of cheese to a crow, as well as cheese, sausages, white and black bread. Perched on a fir tree, a crow was about to have breakfast, but she thought about it: in how many ways can sandwiches be made from these products?
Option 2
a) How many three-digit numbers can be made from the numbers: 0, 2, 4, 6, 8?
b) How many three-digit numbers can be made from the numbers: 0, 2, 4, 6, 8, provided that the numbers should not be repeated?
Count Monte Cristo decided to give Princess Hyde earrings, a necklace and a bracelet. Each piece of jewelry must contain one of the following types of gems: diamonds, rubies or garnets.
a) How many combinations of gemstone jewelry are there?
b) How many jewelry options are there if the earrings must be diamond?
c) How many jewelry options are there if the earrings should be diamond and the bracelet garnet?
For breakfast, you can choose a bun, sandwich or gingerbread with coffee or kefir. How many breakfast options can you make?
Homework : No. 974, 975. (by compiling a tree of options and using the multiplication rule)
№ 974 . a) How many three-digit numbers can be formed from the numbers 0, 2, 4?
b) How many three-digit numbers can be made from the numbers 0, 2, 4, provided that the numbers should not be repeated?
№ 975 . a) How many three-digit numbers can be made from the numbers 1.3, 5.7?
b) How many three-digit numbers can be made from the numbers 1.3, 5.7, provided. What numbers should not be repeated?
Problem numbers are taken from the textbook
"Mathematics-5", I.I. Zubareva, A.G. Mordkovich, 2004.
The purpose of the lesson:
- Introduce the concept of certain, impossible and random events.
- To form knowledge and skills to determine the type of events.
- Develop: computational skill; Attention; the ability to analyze, reason, draw conclusions; group work skills.
During the classes
1) Organizational moment.
Interactive exercise: children must solve examples and decipher words, according to the results they are divided into groups (reliable, impossible and random) and determine the topic of the lesson.
1 card.
| 0,5 | 1,6 | 12,6 | 5,2 | 7,5 | 8 | 5,2 | 2,08 | 0,5 | 9,54 | 1,6 |
2 card
| 0,5 | 2,1 | 14,5 | 1,9 | 2,1 | 20,4 | 14 | 1,6 | 5,08 | 8,94 | 14 |
3 card
| 5 | 2,4 | 6,7 | 4,7 | 8,1 | 18 | 40 | 9,54 | 0,78 |
2) Actualization of the studied knowledge.
The game "Clap": an even number - clap, an odd number - stand up.
Task: from this series numbers 42, 35, 8, 9, 7, 10, 543, 88, 56, 13, 31, 77, ... determine even and odd.
3) Learning a new topic.
You have cubes on the tables. Let's take a closer look at them. What do you see?
Where are dice used? How?
Group work.
Conducting an experiment.
What predictions can you make when rolling a dice?
First prediction: one of the numbers 1,2,3,4,5 or 6 will fall out.
An event that is sure to occur in a given experience is called reliable.
Second prediction: the number 7 will come up.
Do you think the predicted event will happen or not?
It's impossible!
An event that cannot occur in a given experiment is called impossible.
Third prediction: the number 1 will come up.
Will this event happen?
An event that may or may not occur in a given experience is called random.
4) Consolidation of the studied material.
I. Determine the type of event
-Tomorrow it will snow red.
It will snow heavily tomorrow.
Tomorrow, although it is July, it will snow.
Tomorrow, although it is July, there will be no snow.
Tomorrow it will snow and there will be a blizzard.
II. Add a word to this sentence in such a way that the event becomes impossible.
Kolya received an A in history.
Sasha did not complete a single task on the test.
Oksana Mikhailovna (history teacher) will explain the new topic.
III. Give examples of impossible, random and certain events.
IV. Work according to the textbook (in groups).
Describe the events discussed in the tasks below as certain, impossible or random.
No. 959. Petya conceived a natural number. The event is as follows:
a) an even number is conceived;
b) an odd number is conceived;
c) a number is conceived that is neither even nor odd;
d) a number that is even or odd is conceived.
No. 960. You opened this textbook to any page and chose the first noun that came across. The event is as follows:
a) there is a vowel in the spelling of the chosen word;
b) in the spelling of the chosen word there is a letter “o”;
c) there are no vowels in the spelling of the chosen word;
d) there is a soft sign in the spelling of the selected word.
Solve #961, #964.
Discussion of solved tasks.
5) Reflection.
1. What events did you meet in the lesson?
2. Indicate which of the following events is certain, which is impossible, and which is random:
a) summer holidays will not;
b) the sandwich will fall butter side down;
in) academic year will ever end.
6) Homework:
Come up with two reliable, random and impossible events.
Draw one of them.
An event is the result of a test. What is an event? One ball is drawn at random from the urn. Removing a ball from an urn is a test. The appearance of a ball of a certain color is an event. In the theory of probability, an event is understood as something about which, after a certain moment of time, one and only one of the two can be said. Yes, it happened. No, it didn't happen. The possible outcome of an experiment is called an elementary event, and the set of such outcomes is simply called an event.
Unpredictable events are called random. An event is called random if, under the same conditions, it may or may not occur. Rolling a die will result in a six. I have lottery ticket. After the publication of the results of the lottery draw, the event that interests me - winning a thousand rubles, either occurs or does not occur. Example.
Two events that, under given conditions, can occur simultaneously are called joint, and those that cannot occur simultaneously are called incompatible. A coin is thrown. The appearance of the "coat of arms" excludes the appearance of the inscription. The events “a coat of arms appeared” and “an inscription appeared” are incompatible. Example.
An event that always happens is called certain. An event that cannot happen is called impossible. Suppose, for example, a ball is drawn from an urn containing only black balls. Then the appearance of a black ball is a certain event; the appearance of a white ball is an impossible event. Examples. It won't snow next year. When you roll a die, a seven will come up. These are impossible events. Snow will fall next year. Rolling the die will result in a number less than seven. Daily sunrise. These are real events.
Problem Solving For each of the described events, determine what it is: impossible, certain, or random. 1. Of the 25 students in the class, two celebrate their birthday a) January 30; b) February 30th. 2. A literature textbook is randomly opened and the second word is found on the left page. This word begins: a) with the letter "K"; b) with the letter "b".
3. Today in Sochi the barometer shows normal atmospheric pressure. In this case: a) the water in the pan boiled at a temperature of 80º C; b) when the temperature dropped to -5º C, the water in the puddle froze. 4. Throw two dice: a) 3 points on the first dice, and 5 points on the second; b) the sum of the points on the two dice is equal to 1; c) the sum of the points rolled on the two dice is 13; d) 3 points on both dice; e) the sum of points on two dice is less than 15. Problem solving
5. You opened the book to any page and read the first noun you came across. It turned out that: a) there is a vowel in the spelling of the chosen word; b) in the spelling of the selected word there is a letter "O"; c) there are no vowels in the spelling of the chosen word; d) there is a soft sign in the spelling of the selected word. Problem solving
Just not in the online translator.
The Golden Gate is a symbol of Kyiv, one of the oldest examples of architecture that has survived to our time. The golden gates of Kyiv were built under the famous Kiev prince Yaroslav the Wise in 1164. Initially, they were called Southern and were part of the system of defensive fortifications of the city, practically no different from other guard gates of the city. It was the Southern Gates that the first Russian Metropolitan Hilarion called "Great" in his "Sermon on Law and Grace". After the majestic Hagia Sophia was built, the “Great” gates became the main land entrance to Kyiv from the southwestern side. Realizing their significance, Yaroslav the Wise ordered to build a small church of the Annunciation over the gates in order to pay tribute to the Christian religion that dominated the city and Russia. From that time on, all Russian chronicle sources began to call the South Gates of Kyiv the Golden Gates. The width of the gate was 7.5 m, the passage height was 12 m, and the length was about 25 m.
le sport ce n "est pas seulement des cours de gym. C" est aussi sauter toujours plus haut nager jouer au ballon danser. le sport développé ton corps et aussi ton cerveau. Quand tu prends l "escalier et non pas l" ascenseur tu fais du sport. Quand tu fais une cabane dans un arbre tu fais du sport. Quand tu te bats avec ton frere tu fais du sport. Quand tu cours, parce que tu es en retard a l "ecole, tu fais du sport.
