ALL-RUSSIAN OLYMPIAD FOR SCHOOLCHILDREN
IN MATHEMATICS 2017 – 2018 academic year. YEAR
SCHOOL STAGE. 6TH GRADE
(7 points). In the addition example + + =, write the same number in each square and a different number in the triangle so that the example is correct.
(7 points). At the first stop in empty bus 18 passengers entered. Then at each stop 4 people got off and 6 people got on. How many passengers were on the bus between the fourth and fifth stops?
(7 points). The engraver makes plates with letters. He engraves the same letters behind same time, different - perhaps for different things. He spent 50 minutes together on two signs “FASHION HOUSE” and “ENTRANCE”, and made one sign “INTO THE CHIMNEY” in 35 minutes. How long will it take him to make the “EXIT” sign?
(7 points). From Monday to Wednesday the gnome eats for breakfast semolina porridge, from Thursday to Saturday - rice porridge, and on Sunday he makes himself scrambled eggs. By even numbers The gnome tells the truth, but on odd months he tells lies. On which of the first ten days of August 2016 could he say: “Tomorrow I will eat semolina porridge for breakfast”? Justify your answer.
(7 points). Show how to cut the shape shown in the figure into 5 equal shapes. (Shapes are called equal if they can be combined when overlapping. Shapes can be flipped.)
Maximum score-35 for all completed tasks.
Answers and assessment criteria for grade 6.
Answer . 1+9+1=11.
Verification criteria.
2. Answer . 24 people.
Solution .
Method 1. After each stop, not counting the first, the number of passengers on the bus increases by 2 people. This means that from the second to the fourth stop the number of people increased by 6 people. Those. became 18+6=24 people.
Method 2 . From the second stop, 3·4=12 people got off, and 3·6=18 people got on. Those. there were 18-12+18=24 people on the bus.
Verification criteria .
3.Answer . 20 minutes.
Solution . In the signs HOUSE OF FASHION ENTRANCE and INTO THE CHIMNEY we will separate the letters forming the word EXIT, then what will remain from the first tablet is D, O, M, M, O, D, and from the second - D, M, O. Note that HOUSE MODES ENTRANCE differs from INTO THE CHIMNEY by the letters D, O, M, and in time - by 15 minutes (50-35 = 15). This means that it takes 15 minutes to make the letters D, O, M.
Now we know that when making B CHIMNEY, it took 15 minutes to make the letters D, M, O, i.e. the remaining 35-15=20 minutes were needed to make the letters B, Y, X, O, D.
Verification criteria.
4.Answer . On Tuesday, August 2, on Wednesday, August 3, on Friday, August 5,
Solution . If today is Sunday, Monday or Tuesday, then tomorrow the gnome eats semolina porridge and the phrase turns out to be true. This means that on these days the gnome could say the specified phrase only when such a day falls on an even number. There are two such days: Tuesday, August 2 and Monday, August 8. On the remaining days of the week (from Wednesday to Saturday) the phrase becomes incorrect, and the dwarf could only say it if the number was odd: on Wednesday, August 3 and on Friday, August 5.A complete search solution is possible, when each of the 10 days is indicated, could the gnome say the said phrase on that day, and it is explained why he could or could not.
Verification criteria .
5.
Verification criteria .
squares and one triangle, but one of the diagonals is drawn incorrectly
3 points
The figure is cut into 5 figures, each of which consists of two
squares and one triangle, but both diagonals are drawn incorrectly
2 points
Exercise 1
(7 points) Replace the asterisks with numbers so that the equality becomes true and all nine numbers are different: *** + ** = 1056.
Possible answers
- 984 + 72 = 1056
- 982 + 74 = 1056
- 974 + 82 = 1056
- 972 + 84 = 1056
No further explanation required.
Verification criteria
- Any of the possible answers is given - 7 points.
- An answer in which any two numbers match is given - 2 points.
Task 2
(7 points) From Monday to Wednesday, the gnome eats semolina porridge for breakfast, from Thursday to Saturday - rice porridge, and on Sunday he makes himself scrambled eggs.
On even days of the month the gnome tells the truth, and on odd days he tells lies.
On which of the first ten days of August 2016 could he say: “Tomorrow I will eat semolina porridge for breakfast”? Justify your answer.
Answer
Solution. If today is Sunday, Monday or Tuesday, then tomorrow the gnome eats semolina porridge and the phrase turns out to be true. This means that on these days the gnome could say the specified phrase only when such a day falls on an even number. There are two such days: Tuesday, August 2 and Monday, August 8. On the remaining days of the week (from Wednesday to Saturday) the phrase becomes incorrect, and the dwarf could only say it if the number was odd: on Wednesday, August 3 and on Friday, August 5.
A complete search solution is possible, when for each of the 10 days it is indicated whether the gnome could say the specified phrase on that day, and it is explained why he could or could not.
Verification criteria
- Completely justified decision - 7 points.
- Generally the right decision, basic reasoning has been carried out, but one of the days is not indicated in the answer - 4 points.
- Correct answer with incomplete reasoning - 2–3 points.
- Only correct answer - 1 point.
Task 3
(7 points) The number 20 is written on the board. In one move, you are allowed to either double the number or erase its last digit. Is it possible to get the number 25 in a few moves?
Answer . Can.
Solution
The number 25 can be obtained by erasing the last digit of the number 256, which is a power of two. Thus, the necessary chain of transformations might look like this:
20 → 2 → 4 → 8 → 16 → 32 → 64 → 128 → 256 → 25.
There are other solutions.
Verification criteria
- Incomplete solution (for example, it is indicated that 25 can be obtained from the number 256, but it is not indicated how to obtain 256) - 3 points.
Task 4
(7 points) Show how to cut the figure shown in the figure into 5 equal figures. (Shapes are called equal if they can be combined when overlapping. Shapes can be flipped.)
Answers
Verification criteria.
- The correct answer is 7 points.
- squares and one triangle, but one of the diagonals is drawn incorrectly - 3 points.
- The figure is cut into 5 figures, each of which consists of two
- squares and one triangle, but both diagonals are drawn incorrectly - 2 points.
Task 5
Grandmother has three grandchildren. If a grandson finished the first grade, then the grandmother gave him one book, if he finished the second grade, then the grandmother gave him two books, if the third grade, then three books, etc. Books received as gifts over the years were put on one by the grandchildren shelf. There are currently 23 books on the shelf. It is known that one of the grandchildren is at least two years older than the others. What grade did he graduate from?
Answer . Sixth grade.
Solution
From the conditions of the problem it follows that if a grandson graduated from the second grade, then there are 1 + 2 = 3 of his books on the shelf, if he graduated from the third grade, then 1 + 2 + 3 = 6 of his books, etc. For convenience, let's make a table.
Obviously, none of the grandchildren could have completed the seventh grade, since then there would have been at least 28 books on the shelf. Since there are 23 books on the shelf, we need to represent the number 23 as the sum of either two numbers in the second row of the table (for the case when one of the grandchildren has not yet completed first grade), or three such numbers. After a little searching, we find that there are only two options for this representation: 23 = 3 + 10 + 10 = 1 + 1 + 21. In the first case, one grandson graduated from the second grade and two grandchildren graduated from the fourth grade. This contradicts the fact that one of the grandchildren is at least 2 years older than the others.
In the second case, two grandchildren graduated from the first grade and one grandson graduated from the sixth grade, which satisfies all the conditions of the problem. So, the eldest grandson graduated from sixth grade.
Verification criteria.
- Any complete correct solution - 7 points.
- In general, a correct solution with small gaps in justification - 5–6 points.
- The correct answer was received, but the case when there are 3 books on the shelf of the youngest grandson and 10 books for each of the other two grandchildren was not considered and the condition about the oldest grandson was not used - 4 points.
- It was found how many books a grandson has if he completed the second, third, etc. grade, but made no further progress - 1 point.
- Only the answer is given - 0 points.
The maximum score for all completed tasks is 35.
Municipal budget educational institution
"Polaznensky kindergarten №8"
Direct educational activities
Integration with educational areas:
cognition, Physical Culture, health, socialization, communication, artistic creativity, reading fiction, music
TOPIC: "Gnome's Helpers"
Teacher Ponomareva I.G.
"Polaznensky kindergarten No. 8"
TARGET: Formation of children's skills to navigate time using the example of familiarization with days and weeks.
Tasks:
Educational area"Cognition":
Expand your understanding of the days of the week. Introduce the new day of the week - Wednesday.
Strengthen ordinal counting up to 5, the ability to correlate the number of objects with a number.
Practice selecting the necessary geometric shapes (triangle, circle, square, rectangle) based on three criteria: shape, color, size. Strengthen sensory standards
Train children to guess riddles by ear and using models.
Develop visual and auditory perception, memory, attention, thinking.
Teach children to solve a problem and find a way out of the situation together.
Educational field "Physical education":
To formulate children's need for physical activity.
Educational field "Health":
Maintain and improve children's health through eye exercises.
Educational field "Artistic creativity":
Strengthen children's skills in painting ready-made shapes.
Educational field "Music":
Encourage children to participate in the game, perform movements in accordance with the musical and verbal accompaniment.
Educational field "Socialization":
Introduce children to basic rules relationships with peers and adults, the ability to work in a team.
Teach children to work in pairs when completing tasks, and to complete tasks together.
Develop interest in the proposed tasks, a desire to help the character.
Educational field "Communication"
Improve dialogical speech: learn to participate in a conversation, answer questions clearly.
Replenish and activate children's vocabulary by deepening knowledge about the days of the week.
Educational area "Reading fiction":
Introduce children to verbal creativity (poems, riddles, finger games)
Developmental environment:
The umbrella is bright; game didactic manual "Days of the week" (house + gnomes); five sets of Dienesh Blocks; model-coding with figures; mystery model "Kindergarten"; a set of numbers from 1 to 5; silhouettes of the number -3, different sizes, white;
paints, brushes, jars of water, coasters; set of geometric shapes different color; tray; tape recorder with music recording. games "Neighbors"; box with gifts for children (coloring books);
Preliminary work with kids:
Conversation about the days of the week; telling riddles using models and words; familiarity with numbers from 1-5; counting to 5; working with Dienesh blocks; (according to the scheme); games with geometric shapes; painting finished forms;
Motivational and orientation stage:
The teacher has a beautiful umbrella in his hands
Look, guys, I have a beautiful umbrella in my hands, and it’s also magical. You can use it to make different trips.
Do you like to travel?
Want to take a trip?
I know the magic words that will help us go on a journey. Everyone stand under the umbrella.
*The teacher says the words:
"Let's stamp our feet, clap our hands,
Let's turn around ourselves, close our eyes,
Let's say "AH" and we'll be a guest"
Closes the umbrella.
*The teacher draws attention to beautiful house:
Look how beautiful the house is.
Leads the children to the house.
MAIN STAGE:
Is this house familiar to you?
Who lives in that house?
Here is a beautiful house, and he lives in it for a week.
There are exactly seven days in a week, everyone needs to know about this!
The dwarf brothers live here, sing songs together,
And telling fairy tales, the colors change every day.
Guys, who are we visiting?
Let's name the colors of the rooms where the gnomes live.
The week has begun, I got to work...
The week starts on Monday.
(show Gnome -Monday, light green)
On Monday the first gnome in green color painted the house and what color is it?
* (Place Monday Gnome in the house of the week)
After Monday comes Tuesday.
And on Tuesday the blue Dwarf Blue colour painted the house.
What color did Tuesday paint the house? -Which Dwarf did he settle next to?
*(put Tuesday Gnome into the house)
Guys, today is the third day of the week. And it's called - WEDNESDAY. Gnome - Wednesday loves everything orange and he dressed in orange clothes. Wednesday settle next to Tuesday. This is the third day of the week.
Gnome Wednesday wanted to go into his house, but somehow he couldn’t do anything. To enter the house, he needs to complete tasks. But they are too complicated for him. What to do? Shall we help the Gnome?
Aren't you afraid of difficulties? -Well, then go ahead.
1 TASK:
" Help the Dwarf tell me what he's wearing?"
1.And in frost and cold, he will always save you.
To prevent your health from being shaky, put it on your head...
2.With buttons, and pocket, with arms and cuff
He sits on me all day, and night will come,
Curls up and sleeps...?
3. The sleeves are longer than the arms,
So you're wearing...?
4. To put on your legs to jump and gallop.
To prevent your feet from swelling, what should you wear...?
Gnome Wednesday got dressed and went for a walk. And off we went. Choose your friend (choice in pairs).
But the path is not visible. (problem) What to do?
2 TASK:
"Help Wednesday the Gnome lay out the path.
It consists of geometric shapes.
(showing the track diagram)
let's call it (Emphasis on shape, color, size)
We lay out the path from Dienesh blocks according to the pattern.
(standing at the tables)
Well done! The path was laid out correctly and beautifully.
Let's go along the path... (in pairs)
Where have we ended up?
There is a floor, walls, ceiling, roof, windows, doors...
Moms and dads bring their children to this place. Children play here, eat, sing..., sleep, walk and everyone lives together. What is this?
3 TASK:
"Let's tell the Gnome what children do in kindergarten in different days"
On Monday in order, children do exercises.
How many children do exercises? (picture)
And on Tuesday, the whole crowd of children builds a big house.
And then they go for a walk...1,2,3,4,5. (picture)
Count how many children are building the house. What number
Let's denote this number of children?
How many children went for a walk? What number will we use to denote this quantity?
On Wednesday they also don’t sit, study and sleep.
(picture)
How many children are involved? Let's count. What number do we denote?
How many children went to bed? Let's count. What number do we denote?
Well done, you told the Gnome well.
Children in kindergarten love to play with numbers. And everyone has a favorite number.
And the Dwarf does not know what the favorite number is for him and his Dwarf friends.
4 TASK:
"Let's help the Gnome find out his favorite number"
Monday, what day of the week?
First, what is his favorite number?
Tuesday, what day in the week?
So what is his favorite number?
What day is Wednesday?
So his favorite number is the number....
Guys, gnomes live in a house together, next to each other. So they are neighbors.
Let's show Gnome Wednesday how neighbors live.
MUSIC-DID. game "NEIGHBORS"
The gnome enjoyed playing with you so much. He wants to play some more.
Let's play with him in an unusual way, with his eyes.
The gnome will hide, and we will look for it with our eyes.
GYMNASTICS FOR THE EYES
"Our Dwarf jumped up,
Suddenly he sat down.
I ran right, I ran left...
Suddenly disappeared completely (we close our eyes)
Appeared again (eyes opened)
They blinked their eyes and stroked them lightly..."
Guys, let's make a surprise for Wednesday Gnome.
Let's give him his favorite number.
What is the Gnome's favorite number?
Choose who wants to give which... They are all different in size.
Do you think these numbers might please the Gnome?
No, because they are colorless and ugly.
What will we do to make them like you? What is the Gnome's favorite color to paint the numbers in?
SURPRISE FOR THE GNOME
"Color the gift"
Well done! We tried! Place it on a tray and we'll give it to the Gnome.
Guys, Monday and Tuesday also prepared a gift for Wednesday - they baked cookies.
It looks like geometric figures.
Tell the Gnome.
What color are the cookies?
5 TASK:
"Help the Gnome choose cookies"
The gnome likes cookies that are orange and not round.
Place cookies on a tray
Feed the Gnome.
Approaching the House of the Week
Well, Gnome Wednesday, we completed all the tasks. But for some reason the door still doesn’t open. There is a lock on the door. We need to open it.
What do we do? -Can we open it? Let's help the Gnome.
Finger game "There's a lock on the door"
The door opened. And our Gnome Wednesday came home. He says "thank you" to us.
Guys, gnomes change every day each other one by one. And they never get confused. Do you know why?
Because they have their own special hours. They are colored. What part of the clock do you think the hand will be on on Wednesday? (I rearrange the hand on the clock)
Well done to you and me. We completed all the tasks and helped Gnome Wednesday. He gives us a gift (a beautiful box with coloring books in it).
We'll see when we return to kindergarten.
-We take an umbrella and return to the nursery. garden
Reflexively evaluation stage:
Did you enjoy the trip?
Where were you? What day of the week did you meet?
What did you like most?
What was easy?
Which tasks were difficult?
Do you think we completed the tasks?
Why did you do everything?
Did we please the gnome?
And you made me happy today too.
So who are you today? (well done)
Let's clap for each other.
And the guests will clap for us.
Now let's see what gift the Dwarf has prepared for us.
(I open the box, the children take apart the coloring books and sit down to color)
Independent activity children - coloring donated coloring books
Follow-up work
The gnomes were real heroes, handsome men and hard workers.
Of course, everyone had their weaknesses.....
Monday- loved to sleep.
Tuesday- I loved to eat even more.
Wednesday- constantly lifted up.... he lifted up his shirt, pants in front and behind.
Thursday- he constantly picked his nose and tried to pick someone else’s.
Friday- he sneezed endlessly, he sneezed left and right, on everything and everyone.
Saturday- always sticking his nose where it doesn’t belong.
A Sunday- hovered in the clouds and caught flies.
But most of the time they worked, mining gold and precious stones.
They did all this for the sake of one... only woman - beautiful Snow Whites!
(comes out to the music of “royal fanfare”)
They all loved her very much, looked after her and vying with each other to give her compliments.
She responded to them with care and affection... and the gnomes did not miss the opportunity to pamper Snow White.
Monday he sat her lovingly on his lap.
Tuesday massaged her shoulders.
Wednesday gently stroked her head and admired her wonderful hair.
Thursday kissed her white hands.
Friday massaged her tired legs.
Saturday sang romances to her.
A Sunday brushed away flies.
(Mysterious) N
oh they had one more thing favorite hobby which they all did together...
and then Snow Whitewas the most happy woman all over the world.....
BECAUSE....... I REALLY LOVED................DANCE!!!
Snow White And gnomes dance, invite the audience.
