one. . 19 math games. S, - Petersburg: Soyuz, 1999.
2. Mathematical circles in school grades 5-8: Methodological manual for the preparation and conduct of classes in the school mathematics circle. - Moscow: "IRIS - PRESS", 2005.
3. Problems for children from 5 to 15 years old. Collection of tasks for the development of a culture of thinking. - Astana: "Daryn", 2008.
4. . School Olympiad in Mathematics. Tasks and solutions.
- Moscow: "Russian Word", 2004.
5. Yu. Nesterenko, S. Olekhnik, M. Potapov. The best tasks for ingenuity. Moscow: AST - PRESS, 1999.
6.. Mathematics in puzzles, crosswords, teawords, cryptograms, grade 5. - Moscow: School Press, 2002.
7. Republican SPC "Daryn". Problems of the I Republican mathematical tournament of junior schoolchildren "Bastau" (June 15-18, 2008) - Astana, 2009.
eight. . Problems of increased difficulty in the course of mathematics of grades 4-5. Book for the teacher. - Moscow, "Education", 1986.
6. Application.
Educational-methodical complex of the course
Annex 1
Appendix 1.1
Arithmetic operations on natural numbers, zero and their properties
Sweet cherry
In the grocery store 141 kg of cherries in boxes of 10 kg and 13 kg.
How many boxes were brought?
Solution.
Let in thirteen-kilogram boxes but kg of cherries, and in ten-kilogram - b kg.
The numbers but and b - natural. Then the number b is divisible by 10, that is, it ends with the digit 0, and, therefore, the number but ends with the number 1, which means that the number of thirteen-kilogram boxes ends with the number 7, but 13 · 17 = 221, 221> 141, since 13 · 7 = 91, 91 <141.
Thus, there were 7 thirteen-kilogram and 5 ten-kilogram boxes, because = 50.
Answer: 7 boxes of 13 kg and 5 boxes of 10 kg.
On the farm
On the farm, rabbits and pheasants are raised. Currently, there are so many that together 740 heads and 1980 legs.
How many rabbits and pheasants are currently on the farm?
Solution.
Let be NS - the number of pheasants, at - the number of rabbits.
Then 2NS + 4at = 1980 and
NS + at = 740,
where NS = 490, at = 250.
Answer. The farm has 490 pheasants and 250 rabbits.
Numbers from the table
Can you choose 5 numbers from the table, the sum of which is 20?
Solution: All numbers in the table are odd, and the sum of five odd numbers is odd and therefore cannot be equal to 20.
Answer. It is forbidden.
Dragon
The Serpent Gorynych has 2000 heads. The fabulous hero cuts off 1, 17, 21 or 33 heads with one blow, but at the same time, 10, 14, 0 or 48 heads, respectively, grow. If all heads are cut off, then new ones do not grow back.
Will the bogatyr be able to defeat the Serpent Gorynych?
Solution.
The following tactics can be proposed for chopping off the heads of the Gorynych snake:
1) first, we will chop off 21 heads (94 times), new heads will not grow, and the Serpent will have 26 heads;
2) then we will chop off 17 heads three times (recall that this grows to 14 heads) - after which it will remain to chop off 17 heads;
3) chop off 17 heads with the last blow.
(2· · = 0.
Answer. The hero will be able to defeat the Serpent Gorynych.
Grasshopper
The grasshopper jumps in a straight line: the first jump is 1 cm, the second is 2 cm, the third is 3 cm, and so on. Can he, after the twenty-fifth jump, return to the point from which he started?
Solution.
Let the grasshopper jump along the number line and start from the point with the coordinate 0. After the 25th jump, he will be at the point with the odd coordinate (among the numbers from 1 to 25 - odd - an odd number). Since 0 is an even number, it cannot return.
Answer: After the twenty-fifth jump, the grasshopper cannot return to the point from which it started.
The mystery of the ancient manuscript
An ancient manuscript describes a city located on 8 islands. The islands are connected to each other and to the mainland by bridges. 5 bridges go to the mainland; 4 bridges start on 4 islands, 3 bridges start on 3 islands and only one bridge can be passed to one island.
Could there be such an arrangement of bridges?
Solution.
Find the number of ends for all bridges:
5 + 4 · 4 + 3 · 3 + 1 = 31.
31 is an odd number.
Since the number of ends of all bridges must be even, there cannot be such an arrangement of bridges.
Answer: There cannot be such an arrangement of bridges.
Appendix 1.2
Divisibility of natural numbers
For training
Among the four statements:
"number but divisible by 2 ″, “number but is divisible by 4 ″, “number but is divisible by 12 ″, “number but divisible by 24 ″ - three true and one false.
Which?
Answer.
Note that “the number but divisible by 24 ″ ⇒ “number but divisible by 12 ″ ⇒ “number but divisible by 4 ″ ⇒ “number but is divisible by 2 ″. Therefore, only the statement “the number but is divisible by 24 ″.
Lucky tickets
Bus tickets have numbers from 000001 to 999999. A ticket is called lucky if the sum of the first three digits is equal to the sum of the last three.
Prove that the sum of all lucky ticket numbers is divisible by 9, 13, 37, and 1001.
Proof.Lucky ticket with a number but1but2but3but4but5but6 corresponds to the only lucky ticket with a number b1b2b3b4b5b6 such that
but1 + b1 = 9;
but2 + b2 = 9;
…
but6 + b6 = 9.
Therefore, the sum of all the lucky ticket numbers is divisible by and, therefore, by 9, 13, 37 and 1001.
Ch. Etc.
In the wild west
Cowboy Joe entered the bar. He bought a bottle of whiskey for $ 3, a pipe for $ 6, three packs of tobacco, and nine boxes of waterproof matches. The bartender said, "You get $ 11 80 for everything." Joe drew his revolver instead of answering.
Why did he think the bartender was going to cheat him?
Answer: It follows from the condition that the total cost of the entire purchase should be divisible by 3, and $ 11.8 is not divisible by 3.
The case in the savings bank
Is it possible to change 25 rubles with ten bills of 1, 3 and 5 rubles?
Answer: It is forbidden. And not at all because such bills do not exist. The sum of an even number of odd terms cannot be an odd number.
Lost weight
The set included 23 weights weighing 1 kg, 2 kg, 3 kg,… 23 kg.
Is it possible to decompose them into two equal parts according to the mass of the pile, if you have lost a weight of 21 kg?
Solution.
Number S = (1 + 23) + (2 + 22) +… + (11 + 13) + 12 - even.
Consequently, (S - 21) cannot be decomposed into two piles of equal weight.
Answer: It is impossible to decompose weights weighing 1 kg, 2 kg, 3 kg, ... 23 kg into two equal parts by the mass of a pile, if a weight of 21 kg has been lost.
Appendix 1.3
Problems using GCD and LCM
Find the remainder
When divided by 2, a number gives a remainder of 1, and when divided by 3, a remainder of 2.
What is the remainder of this number when divided by 6?
Solution.
Since when dividing an integer by 6 you can get one of the remainders: 0, 1, 2, 3, 4 and 5, the set of non-negative integers can be divided into disjoint subsets of numbers of the form 6k, 6k + 1, 6k + 2,
6at + 3, 6k + 4 and 6at + 5, where k = 0, 1, 2, 3, … .
Since when divided by 2 this number gives a remainder of 1, then it is odd, so it remains to consider numbers of the form 6k + 1, 6at + 3 and 6at + 5.
Numbers like 6k + 1 when divided by 3 gives a remainder of 1, numbers like 6k + 3 are multiples of 3 and only numbers of the form 6k + 5 divided by 3 gives a remainder of 2.
Therefore, the number has the form 6at + 5, that is, dividing by 6 gives a remainder of 5.
Answer.
If, when divided by 2, a number gives a remainder of 1, and when divided by 3, a remainder of 2, then when divided by 6, the number gives a remainder of 5.
Appendix 1.4
Tasks and puzzles
I. Oral work
1. You are a bus driver. The bus originally had 23 passengers. At the first stop, 3 women got off and 5 men got in. At the second stop, 4 men entered and 7 women exited. How old is the driver?
2. Selling a parrot in the store, the seller promised that the parrot would repeat every word he heard. The buyer was very happy, but when he came home, he found that the parrot was "mute as a fish." However, the seller did not lie. How could this be?
3. Petya decided to buy ice cream for Masha, but 30 tons were not enough for him, and only 10 tons for Masha. Then they decided to add up their money, but again 10 tons were not enough to buy even one ice cream. How much did a serving of ice cream cost? How much money did Petya have?
II. Learning new material
1. I thought of a number, multiplied it by two, added three and got 17. What number am I thinking?
2. Once the devil offered a loafer to earn money.“As soon as you cross this bridge,” he said, “your money will double. You can cross it as many times as you want, but after each crossing, give me 24 tons for it. " The slacker agreed and ... after the third passage he was left penniless. How much money did he have at first?
3. Three boys each have a certain number of apples. The first boy gives the others as many apples as each of them has. Then the second boy gives the other two as many apples as each of them now has; in turn, the third gives each of the other two as much as each has at that moment. After that, each of the boys, it turns out, has 8 apples. How many apples did each boy have at the beginning?
4. Solve the puzzles: a) * * b) * * c) D R A M A
* * * D R A M A
* * 8 * 9 8 T E A T R
III. Homework
1. Geese were flying over the lakes. On each lake half of the geese sat down and half a goose, the rest flew further. All sat down on seven lakes. How many geese were there?
( Half a goose cannot land, therefore, a whole number of geese landed on each lake.)
2. Solve the rebus: K O K A
C O L A
V O D A
Rebus
Answer: two
Answer: diagonal Answer: diameter
Answer: fraction
WISE THOUGHTS
“A person is like a fraction: in the denominator - what he thinks of himself, in the numerator - what he really is. The larger the denominator, the smaller the fraction. "
Lev Tolstoy
Answer: numerator
Answer: a challenge.
Answer: ruler
Answer: minus
Answer: line segment
Answer: degree
WISE THOUGHTS
“Knowledge is the most excellent of possessions. Everyone strives for it, but it itself does not come. "
Al-Biruni
Number puzzles
It is required to decipher the notation of arithmetic equality, in which the numbers are replaced by letters, and different numbers are replaced by different letters, the same - the same. It is assumed that the original equality is correct and written according to the usual rules of arithmetic. In particular, in the notation of a number, the first digit from the left is not the digit 0; the decimal number system is used.
Addition
# 1. Livestock rebus
B + B E E E = M U U U
Solution: Since when adding these numbers, the digit E in the tens place changed to the digit Y, the sum of the single-digit numbers B and E is a two-digit number starting with one. Since, in addition to increasing the number in the tens place by one, the number in the hundreds place has also changed, then E = 9, B = 1, Y = 0.
Answer: 1 + 1999 = 2000.
No. 2. Coca Cola
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+ |
TO |
O |
TO |
BUT |
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TO |
O |
L |
BUT |
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IN |
O |
D |
BUT |
No. 3. Drama
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+ |
Have |
D |
BUT |
R |
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Have |
D |
BUT |
R |
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D |
R |
BUT |
M |
BUT |
No. 4. Cross
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+ |
WITH |
NS |
O |
R |
T |
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WITH |
NS |
O |
R |
T |
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TO |
R |
O |
WITH |
WITH |
No. 5. Dogs
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+ |
B |
BUT |
R |
B |
O |
WITH |
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B |
O |
B |
AND |
TO |
||
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WITH |
O |
B |
BUT |
TO |
AND |
No. 6. friendship
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+ |
BUT |
H |
D |
R |
E |
Th |
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F |
BUT |
H |
H |
BUT |
||
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D |
R |
Have |
F |
B |
BUT |
No. 7. Milk
| Due to the large volume, this material is located on several pages: 1 2 3 4 5 6 7 |
