In this lesson you will review everything you know about arithmetic operations. You already know four arithmetic operations: addition, subtraction, multiplication, division. Also in this lesson we will look at all the rules associated with them and how to check calculations. You will learn about the properties of addition and multiplication, and consider special cases of various arithmetic operations.
Addition is indicated by a “+” sign. An expression in which numbers are connected by a “+” sign is called a sum. Each number has a name: the first term, the second term. If we perform the addition action, we get the value of the sum.
For example, in the expression:
This is the first term, this is the second term.
This means that the value of the sum is .
Let us recall special cases of addition with the number 0:
If one of the two terms is equal to zero, then the sum is equal to the other term.
Find the value of the sum:
Solution
If one of the two terms is equal to zero, then the sum is equal to the other term, so we get:
1. 
2. 
Answer: 1. 237; 2.541.
Let us repeat two properties of addition.
Commutative property of addition: Rearranging the terms does not change the sum.
For example:
Combinative property of addition: two adjacent terms can be replaced by their sum.
For example:
Using these two properties, terms can be rearranged and grouped in any way.
Calculate in a convenient way:
Solution
Let's consider the terms of this expression. Let's determine whether there are any that when added together will result in a round number.
Let's use the commutative property of addition - rearrange the second and third terms.
Let's use the grouping of the first and second terms, the third and fourth terms.
Answer: 130.
Subtraction is indicated by a “-” sign. Numbers connected by a minus sign form a difference.
Each number has a name. The number from which it is subtracted is called the minuend. The number that is being subtracted is called the subtrahend.
If we perform the subtraction action, we get the difference value.
If one of the two factors equal to one, then the value of the product is equal to another factor.
If one of the factors is zero, then the value of the product is zero.
If you subtract zero from a number, you get the number from which you subtracted.
If the minuend and the subtrahend are equal, then the difference is zero.
Calculate in a convenient way:
Solution
In the first expression, zero is subtracted from the number. Accordingly, you get the number from which you subtracted.
1. 
In the second expression, the minuend and subtrahend are equal, respectively, the difference is zero.
2. 
Answer: 1. 1864; 20.
It is known that addition and subtraction are mutually inverse operations.
Check the calculations:
1. 
2. 
Solution
Let's check whether the addition was performed correctly. It is known that if you subtract the value of one of the terms from the value of the sum, you get another term. Subtract the first term from the sum:
Let's compare the result obtained with the second term. The numbers are the same. This means that the calculations were performed correctly.
It was also possible to subtract the second term from the sum value.
Let's compare the result obtained with the first term. The numbers are equal, which means the calculations were done correctly.
Let's check if the subtraction was performed correctly. It is known that if you add the subtrahend to the difference value, you get the minuend. Let's add the subtrahend to the difference value:
The result obtained and the minuend coincide, that is, the subtraction was performed correctly.
There is another way to check. If you subtract the difference value from the minuend, you get the subtrahend. Let's check the subtraction in the second way.
The result obtained coincides with the subtracted one, which means that the difference value was found correctly.
Answer: 1. true; 2. true.
To indicate the action of multiplication, two symbols are used: “”, “”. Numbers connected by a multiplication sign form a product.
Each number has a name: the first factor, the second factor.
For example:
In this case, this is the first multiplier, and this is the second multiplier.
It is also known that multiplication replaces the sum of identical terms.
The first factor shows which term is repeated. The second factor shows how many times this term is repeated.
If we perform the multiplication action, we get the value of the product.
Find the meaning of the expressions:
Solution
Let's look at the first piece. The first factor is equal to one, which means the product is equal to the other factor.
Let's look at the second piece. The second factor is zero, which means the value of the product is zero.
Answer: 1. 365; 20.
Commutative property of multiplication.
Rearranging the factors does not change the product.
Combinative property of multiplication.
Two adjacent factors can be replaced by their product.
Using these two properties, factors can be rearranged and grouped in any number of ways.
Distributive property of multiplication.
When multiplying a sum by a number, you can multiply each term separately by it and add the resulting results.
Calculate in a convenient way:
Solution
Let's take a closer look at the multipliers. Let's determine if there are any that, when multiplied, produce a round number.
Let's use a permutation of factors and then group them.
Answer: 2100.
The following symbols are used to indicate the division action:
Numbers connected by a division sign form a quotient. The first number in the record - the one that is being divided - is called the dividend. The second number in the notation - the one that is being divided by - is called the divisor.
If we perform the division operation, we get the value of the quotient.
Multiplication and division are reciprocal operations.
Check the calculations:
2. 
Solution
It is known that if the value of a product is divided by one of the factors, the second factor is obtained.
To check the correctness of the multiplication, divide the product by the first factor.
The result obtained coincides with the second factor, which means the multiplication was performed correctly.
You can also divide the value of the product by the second factor.
The resulting quotient value coincides with the value of the first factor. This means that the multiplication was performed correctly.
Let's check the correctness of division by multiplication. If you multiply the value of a quotient by a divisor, you get the dividend.
Let's multiply the value of the quotient by the divisor.
Let's compare the result with the divisor. The numbers match, which means the division was done correctly.
The result of division can be checked in another way.
If you divide the dividend by the quotient, you get a divisor.
The result is the same as the divisor. This means that the division is done correctly.
Answer: 1. true; 2. true.
If zero is divided by any other number, the result is zero.
You cannot divide by zero.
If you divide a number by 1, you get the number that was divided.
If the dividend and the divisor are equal, then the quotient is equal to one.
In this lesson we recalled the following arithmetic operations: addition, subtraction, multiplication, division. We also repeated various properties of these actions and special cases associated with them.
Bibliography
- Volkova. S.I. Mathematics. Test work 4th grade for the textbook Moro M.I., Volkova S.I. 2011. - M.: Education, 2011.
- Moro M.I. Mathematics. 4th grade. In 2 parts. Part 1. - M.: Education, 2011.
- Moro M.I. Mathematics. 4th grade. In 2 parts. Part 2. - M.: Education, 2011.
- Rudnitskaya V.N. Mathematics tests. 4th grade. To the textbook Moro M.I. 2011. - M.: Exam, 2011.
- Mat-zadachi.ru ().
- Videouroki.net ().
- Festival.1september.ru ().
Homework
- Textbook: Volkova. S.I. Mathematics. Test work 4th grade for the textbook Moro M.I., Volkova S.I. 2011. - M.: Education, 2011.
- Test work No. 1 Option 1 p. 6.
- Textbook: Rudnitskaya V.N. Mathematics tests. 4th grade. To the textbook Moro M.I. 2011. - M.: Exam, 2011.
- Ex. 11 page 9.
Sections: Mathematics
Class: 6
Lesson Objectives:
1. Educational: repetition, generalization and testing of knowledge on the topic: “Divisibility of natural numbers”; development of basic skills.
2. Developmental: develop students’ attention, perseverance, perseverance, logical thinking, mathematical speech.
3. Educational: through the lesson, cultivate an attentive attitude towards each other, instill the ability to listen to comrades, mutual assistance, and independence.
Lesson objectives:
To develop the ability to apply the concept of divisors and multiples; develop thinking and elements creative activity; apply divisibility criteria in the simplest situations; finding GCD and LCM numbers, developing observation and logical thinking.
Lesson type– combined.
Lesson form– lesson with computer support.
Equipment:
1. Board and chalk.
2. Computer and projector.
3. Paper version of all tasks.
During the classes.
Numbers rule the world.
Pythagoras.
1. Organizational moment.
2. Communicate the purpose of the lesson.
3. Updating of basic knowledge.
1. What is a number divisor? A?
2. What is a multiple of a number? A?
3. Is there a greatest multiple?
4. Formulate the signs of divisibility?
5. Which numbers are called prime and which are composite?
(Students’ report about Pythagoras, Eratosthenes, Euclid)
Historical information:
| Euclid - ancient Greek scientist (365 - 300 BC). Very little is known about the life of this great scientist. He lived and worked in Alexandria, the city founded by Alexander the Great. Many legends are associated with the name of Euclid. One of them says that King Ptolemy asked Euclid: “Is there a shorter way to knowledge of geometry?”, to which the scientist replied: “There is no royal road to geometry!” Euclid studied number theory a lot: it was he who proved that there are infinitely many prime numbers. The algorithm for finding the gcd of two numbers is called the Euclidean algorithm. The ancient Greek mathematician Euclid, in his book Elements, which was the main textbook in mathematics for two thousand years, proved that there are infinitely many prime numbers, i.e. Behind every prime number there is an even prime number. |
| Pythagoras (6th century BC) and his students studied the question of the divisibility of numbers. They called a number equal to the sum of all its divisors (without the number itself) a perfect number. For example, the number 6 (6 = 1 + 2 + 3), 28 (28 = 1 + 2 + 4 + 7 + 14) are perfect. The following perfect numbers are 496, 8128, 33550336 The Pythagoreans only knew the first three perfect numbers. The fourth 8128 became known in the 1st century BC. The fifth number, 33550336, was found in the 15th century. By 1983, 27 perfect numbers were already known. But scientists still don’t know whether there is an odd perfect number or whether there is a largest perfect number. The interest of ancient mathematicians in prime numbers stems from the fact that any natural number greater than 1 is either a prime number or can be composed as a product of prime numbers: 14 = 2∙7, 16 = 2∙2 ∙2∙2 The question arises: is there a last (largest) prime number? |
Task: A prime number has been conceived. The next natural number is also prime. What numbers are we talking about?
Answer: 2.3.
6. What numbers are called relatively prime?
7. Explain how to find GCD (LCD) of two numbers.
(Student’s message about finding the gcd of two numbers)
One day the numbers 24 and 60 argued about how to find a gcd. The number 24 stated that you first need to find among all the divisors total numbers, and then choose the largest number from them. And number 60 objected:
- Well, what are you talking about! I don't like this method. I have too many divisors, and in listing them I might miss one. What if it turns out to be the largest? No, I don't like this method. And they decided to turn to the Master of Business Sciences for help. And the master answered them:
- Yes, 24, your method of finding gcd of numbers can be used, but it is not always convenient. But you can find GCD in a different way.
You need to factor 24 and 60 into prime factors.
| 24 | 2 |
| 12 | 2 |
| 6 | 2 |
| 3 | 3 |
| 1 |
| 60 | 2 |
| 30 | 2 |
| 15 | 3 |
| 5 | 5 |
| 1 |
24 = 2³ ∙ 3
60 = 2² ∙ 3 ∙ 5
You need to take the common divisors of numbers with a smaller exponent.
GCD (24;60) = 2² ∙ 3 = 12.
And to find the LCM of two numbers you need:
- Factor into prime factors;
- Write down all the prime factors that are included in the first number and in the second number with the largest exponent.
Means:
24 = 2³ ∙ 3 60 = 2² ∙ 3 ∙ 5 NOC (24;60) = 2³∙ 3 ∙ 5 = 120.
Clients have repeatedly come to me and were concerned about one question: why is it that their relationships are different from time to time? Is the same scenario repeating itself? It seems like you act differently, but... still the relationship ends equally unsuccessfully. Like last time, like the time before. After 2-3 attempts, suspicions arise that something is wrong with you. Maybe this is the same bad luck? I don't believe in fate or that anyone is destined to be single. I believe that specific communication problems hinder relationships. Let's identify and change the harmful pattern.
Troubled relationships come with a wide range of problems. These include scandals, mutual claims, misunderstanding, unavailability, discontent, mistrust, narcissism, toxic relationships, psychological and physical violence (abuse), alcohol and drug abuse, etc. and so on. In the end, the couple comes to separation. If this happens once, it’s an accident, an accident. But what if this becomes a constant “rake”?
I don't pretend that I will cover everything possible options. I will tell you about those that come across more often.
Let's start with the first three:
- fear of intimacy
- habit
- Demand/Remove scenario
Fear of intimacy is like a boomerang that comes back
Intimacy in a relationship is emotional closeness to your partner. Allowing your inner guard to relax and put down the weapon. You can openly share your feelings and calmly accept your partner’s feelings, including negative ones. Share your inner world.
If one person in a couple is afraid of intimacy because he has previously been seriously hurt or experienced emotional trauma, then he either rejects intimacy or chooses someone like himself as a partner.
In these cases, the relationship is devoid of warmth and openness. The second person feels kind of like in a couple, but at the same time kind of like being alone. Emotions are a traffic light that shows where to go, so talking about how you feel helps you understand someone else's behavior. If there is neither one nor the other, you can only guess, or... leave. Dissatisfaction with the relationship, either in one of the couple or in both, leads to separation.
What to do?
Intimacy does not appear on its own out of nowhere - above it work. Some people have to work harder and longer than others. Here are some approximate directions:
- make it a rule to express positive emotions about your relationship and your partner. You shouldn’t assume that he already knows why he’s talking. It is necessary to speak, because it is important for everyone to know from the primary source that they are valued, loved and respected.
- create conditions for the opportunity to be together. For some it is important to talk, for others it is important to touch each other, for others it is important to play chess, for others it is important to walk - your choice. The more small children you have, the more important this point is.
- learn to express feelings using I-messages. Do not speak: “Why didn’t you warn me?!” Say it like this: “I’m so upset because I wanted to know about it first.”.
Habitual behavior, including in thoughts
Habit is second nature, have you heard? The same applies to the way we think. Yes, yes, if you think in a certain way for many years in a row, then a habitual pattern will develop, which is the first to work.
Let me give you an example: an hour passed, but my husband still did not respond to the SMS. What are the possible explanations why?
- “What if something happened to him?!”
- “He doesn’t care what I write!”
- “He’s less interested in me than what he’s doing...”
- “He’s probably having fun flirting with someone there again!”
- “He’s at a meeting (on the road, etc.)”
- “He will answer when he can.”
Do you see that each option leads to specific emotions, and those, in turn, lead to actions?
One option will be more familiar to you than the rest. It will work faster and will seem like the real thing. Moreover, every day we automatically do our usual actions a thousand times, so this becomes a thousand firsts.
To react differently feels alien and not true. Even if a person understands that the usual path does not lead to anything positive for both parties, he still continues to choose this particular option.
A habit is formed if the behavior provides a reward or benefit. Example: If breaking dishes provides short-term relief from strong negative emotions, there is a high chance of it happening again. A person throws cups again and again, even if he later feels ashamed and realizes that he shouldn’t have done that.
What to do?
Identify habitual patterns: independently or with the help of a psychotherapist. Try to understand whether there is a benefit involved, and if so, what kind and what to do with it. Systematically work on choosing constructive and satisfying forms of behavior.
Scenario Demand/Withdraw
There is one interesting theory about problematic and toxic scenarios in relationships (Papp, Kouros, Cummings).
Briefly, what is the essence: partners are involved in dialogue according to certain rules, one plays the role of the one who demands, and the second - the one who moves away.
The trap is that the more one partner demands, the more the other withdraws. Noticing this, the demander intensifies his claims and requests, and the distancer increases the distance even more. The picture for illustration is typical: the wife, with her hands raised and a distorted face, is shouting something, and the husband, with his arms crossed on his chest and with a concrete expression on his face, is looking out the window.
The bad news is that the roles in this scenario are set by whoever starts. If he is depressed, then the likelihood of developing the Demand/Withdrawal scenario increases. Insecure people are also quickly drawn into this scenario. People with avoidant personality traits or with an avoidant attachment style react more strongly in the withdrawal pattern. The more angry their partner is at them, the more distance they take.
The distribution of power in a couple also influences: how fewer decisions accepts one partner, the less opportunity he has to participate in the life of the couple, the higher the likelihood that he will take on a demanding role and his demands will be high.
It happens that the script appears only in certain topics: habits, sexual preferences, mutual promises, personality and character. Sometimes it manifests itself in conversations about money.
What to do?
Be aware of the existence of the script. When he appears, try to stop: either stop demanding, or stop moving away. There are more constructive ways to interact.
Compiled by teacher of the department of higher mathematics Ishchanov T.R.
Lesson No. 1. Elements of combinatorics
Theory.
Multiplication rule: if from a certain finite set the first object (element) can be selected in ways, and the second object (element) - in ways, then both objects ( and ) in the specified order can be selected in ways.
Addition rule: if some object can be selected in ways, and an object can be selected in ways, and the first and second methods do not intersect, then any of the objects ( or ) can be selected in ways.
Practical material.
1.(6.1.44. L) How many different three-digit numbers can be made from the numbers 0, 1, 2, 3, 4 if:
a) numbers cannot be repeated;
b) numbers may be repeated;
c) the numbers must be even (numbers may be repeated);
d) the number must be divisible by 5 (numbers cannot be repeated)
(Answer: a) 48 b) 100 c) 60 d) 12)
2. (6.1.2.) How many numbers containing at least three different digits can be made from the numbers 3, 4, 5, 6, 7? (Answer: 300.)
3. (6.1.39) How many four-digit numbers can be made so that any two adjacent digits are different? (Answer: 6561)
Theory. Let us be given a set consisting of n different elements.
An arrangement of n elements by k elements (0?k?n) is any ordered subset of a given set containing k elements. Two arrangements are different if they differ from each other either in the composition of the elements or in the order in which they appear.
The number of placements of n elements by k is indicated by a symbol and is calculated by the formula:
where n!=1·2·3·…·n, and 1!=1.0!=1.
Practical material.
4. (6.1.9 L.) Make up different arrangements of two elements each from the elements of the set A=(3,4,5) and count their number. (Answer: 6)
5. (6.1.3 L) In how many ways can three prizes be distributed among 16 competitors? (Answer: 3360)
6. (6.1.11. L) How many five-digit numbers are there, all of whose digits are different? Note: take into account the fact that numbers like 02345, 09782, etc. We don’t count them as five-digit. (Answer: 27,216)
7. (6.1.12.L.) In how many ways can a tricolor striped flag (three horizontal stripes) be composed if there is material of 5 different colors? (Answer: 60.)
Theory. A combination of n elements of k elements each (0?k?n) is any subset of a given set that contains k elements.
Any two combinations differ from each other only in the composition of the elements. The number of combinations of n elements by k is indicated by a symbol and is calculated by the formula:
Practical material.
8.(6.1.20.) Make up various combinations of two elements from the elements of the set A=(3,4,5) and count their number. (Answer: 3.)
9. (6.1.25.) A group of tourists from 12 boys and 7 girls chooses by lot 5 people to prepare dinner. How many ways are there in which to get into this “five”:
a) only girls; b) 3 boys and 2 girls;
c) 1 boy and 4 girls; d) 5 young men; e) tourists of the same sex.
(Answer: a) 21; b) 4620; c) 420; d) 792; e) 813.)
Theory. An n-element permutation is an arrangement of n elements by n elements. Thus, to indicate one or another permutation of a given set of n elements means to choose a certain order of these elements. Therefore, any two permutations differ from each other only in the order of the elements.
The number of permutations of n elements is indicated by a symbol and is calculated by the formula:
Practical material.
10.(6.1.14.L) Create various permutations from the elements of the set A=(5;8;9). (Answer: 6)
11.(6.1.15.L) In how many ways can a ten-volume book of D. London’s works be arranged on a bookshelf, arranging them:
a) in any order;
b) so that volumes 1, 5, 9 are side by side (in any order);
c) so that volumes 1, 2, 3 are side by side (in any order).
(Answer: a) 10! b) 8!?3! V) )
12. (1.6.16.L.) There are 7 chairs in the room. In how many ways can 7 guests be accommodated? 3 guests? (Answer: 5040; 210)
Selection scheme with return.
Theory. If an ordered selection of k elements from n elements is returned back, then the resulting selections represent allocations with repetitions. The number of all placements with repetitions of n elements by k is denoted by the symbol and calculated by the formula:
If, when selecting k elements from n, the elements are returned back without subsequent ordering (thus, the same elements can be removed several times, i.e., repeated), then the resulting samples are combinations with repetitions. The number of all combinations with repetitions of n elements in k is denoted by a symbol and calculated by the formula:
Practical material.
13.(6.1.29.) From elements (numbers) 2, 4, 5, make up all arrangements and combinations with repetitions of two elements. (Answer: 9; 6)
14. (6.1.31.L.) Five people entered the elevator on the 1st floor of a nine-story building. In how many ways can passengers exit the elevator at the desired floors? (Answer: 
)
15. (6.1.59.L.) There are 7 types of cakes in the pastry shop. In how many ways can you buy from it: a) 3 cakes of the same type; b) 5 cakes? (Answer: a) 7; b) 462)
Theory. Let there be k in a set of n elements various types elements, while the 1st type of elements is repeated once, the 2nd - once, . . . , k-th time, and . Then permutations of elements of a given set are permutations with repetitions.
The number of permutations with repetitions (sometimes speaks of the number of partitions of a set) of n elements is denoted by a symbol and calculated by the formula:
Practical material.
16.(6.1.32.) How many different “words” (a “word” means any combination of letters) can be made by rearranging the letters in the word AGA? MISSISSIPPI?
Solution.
In general, from three letters you can make various three-letter “words”. In the word AGA, the letter A is repeated, and rearranging identical letters does not change the “word.” Therefore, the number of permutations with repetitions is less than the number of permutations without repetitions by as many times as repeating letters can be rearranged. In this word, two letters (1st and 3rd) are repeated; therefore, so many different permutations of three-letter “words” can be made from the letters of the word AGA: . However, the answer can be obtained more simply: . Using the same formula, we will find the number of eleven-letter “words” when rearranging the letters in the word MISSISSIPPI. Here (4 letters S), (4 letters I), , therefore
17.(6.1.38.L.) How many different permutations of letters are there in the word TRACTATE? And in the “word” AAUUUUUUU? (Answer: 420;210)
SHARE
SHARE
2. Have the ability to be divided by another number without a remainder (mat.). Even numbers are divisible by two.
3. with someone or something. To divide property with someone (legal).
4. than with someone-what. Distributing from one's own, supplying with something from one's property, sharing with someone. He shared his income with us. Sharing your last penny with a friend, 5. pen. When informing, telling someone about something, give someone some of your knowledge and information. Share news with friends. Share knowledge with the masses.
|| By telling something to someone, by confiding in someone (your experiences), to attract sympathy, to a shared experience. Share grief.
Ushakov's Explanatory Dictionary. D.N. Ushakov. 1935-1940.
Antonyms:
See what “SHARE” is in other dictionaries:
Cm … Synonym dictionary
share- possession of power, causation to share impressions causation, knowledge to share information action, indirect object... Verbal compatibility of non-objective names
SHARE, share, share; imperfect 1. (1st person and 2nd person not used). Have the ability to divide by another number without a remainder. Ten is divisible by five. 2. (1st person and 2nd person units not used). To be distributed, to fall apart. Students… … Ozhegov's Explanatory Dictionary
share- share, share, share and obsolete shares; prib. dividing and dividing... Dictionary of difficulties of pronunciation and stress in modern Russian language
share- share generously... Dictionary of Russian Idioms
I nesov. nepereh. 1. Carry out division of property for further separate residence. 2. Use something together with someone. Ott. trans. Tell, tell someone about something, sharing your knowledge with someone. 3. transfer... ... Modern Dictionary Russian language Efremova
Share, sharing, dividing, dividing, dividing, dividing, dividing, dividing, dividing, dividing, dividing, dividing, dividing, dividing, dividing, dividing, dividing, dividing, dividing, dividing, dividing, dividing, dividing, dividing,... ... Word forms
Multiply unite multiply multiply combine unite multiply greedy stingy... Dictionary of antonyms
share- sharing, sharing, sharing... Russian spelling dictionary
Books
- Sharing is good, Brigitte Weninger. As soon as Max the Mouse found a large apple tree in the clearing with juicy red apples hanging on its branches, he firmly decided to share them with his friends. And to collect the fruits and...
- Sharing is good, Brigitte Weninger. As soon as Max the Mouse found a large apple tree in the clearing with juicy red apples hanging on its branches, he firmly decided to share them with his friends. And to organize Apple...