Which action is the first division or multiplication? The order of performing mathematical operations

Several rules apply when multiplying and dividing integers. In this lesson we will look at each of them.

When multiplying and dividing integers, pay attention to the signs of the numbers. It will depend on them which rule to apply. Also, it is necessary to study several laws of multiplication and division. Studying these rules allows you to avoid some annoying mistakes in the future.

Lesson content

Multiplication laws

We looked at some of the laws of mathematics in the lesson. But we have not considered all the laws. There are many laws in mathematics, and it would be wiser to study them sequentially as needed.

First, let's remember what multiplication consists of. Multiplication consists of three parameters: multiplicand, multiplier And works. For example, in the expression 3 × 2 = 6, the number 3 is the multiplicand, the number 2 is the multiplier, and the number 6 is the product.

Multiplicand shows what exactly we are increasing. In our example we increase the number 3.

Factor shows how many times you need to increase the multiplicand. In our example, the multiplier is the number 2. This multiplier shows how many times the multiplicand 3 needs to be increased. That is, during the multiplication operation, the number 3 will be doubled.

Work This is the actual result of the multiplication operation. In our example, the product is the number 6. This product is the result of multiplying 3 by 2.

The expression 3 × 2 can also be understood as the sum of two triplets. Multiplier 2 in this case will show how many times you need to repeat the number 3:

Thus, if the number 3 is repeated twice in a row, the number 6 will be obtained.

Commutative law of multiplication

The multiplicand and the multiplier are called by one common word - factors. The commutative multiplication law is as follows:

Rearranging the places of the factors does not change the product.

Let's check if this is true. For example, let's multiply 3 by 5. Here 3 and 5 are factors.

3 × 5 = 15

Now let's swap the factors:

5 × 3 = 15

In both cases, we get the answer 15, which means we can put an equal sign between the expressions 3 × 5 and 5 × 3, since they are equal to the same value:

3 × 5 = 5 × 3

15 = 15

And with the help of variables, the commutative law of multiplication can be written as follows:

a × b = b × a

Where a And b- factors

Combinative law of multiplication

This law says that if an expression consists of several factors, then the product will not depend on the order of actions.

For example, the expression 3 × 2 × 4 consists of several factors. To calculate it, you can multiply 3 and 2, then multiply the resulting product by the remaining number 4. It will look like this:

3 × 2 × 4 = (3 × 2) × 4 = 6 × 4 = 24

This was the first solution. The second option is to multiply 2 and 4, then multiply the resulting product by the remaining number 3. It will look like this:

3 × 2 × 4 = 3 × (2 × 4) = 3 × 8 = 24

In both cases, we get the answer 24. Therefore, we can put an equal sign between the expressions (3 × 2) × 4 and 3 × (2 × 4), since they are equal to the same value:

(3 × 2) × 4 = 3 × (2 × 4)

and with the help of variables the associative law of multiplication can be written as follows:

a × b × c = (a × b) × c = a × (b × c)

where instead of a, b,c Any numbers can be.

Distributive law of multiplication

The distributive law of multiplication allows you to multiply a sum by a number. To do this, each term of this sum is multiplied by this number, then the resulting results are added.

For example, let's find the value of the expression (2 + 3) × 5

The expression in parentheses is the sum. This sum must be multiplied by the number 5. To do this, each term of this sum, that is, the numbers 2 and 3, must be multiplied by the number 5, then the resulting results must be added:

(2 + 3) × 5 = 2 × 5 + 3 × 5 = 10 + 15 = 25

This means that the value of the expression (2 + 3) × 5 is 25.

Using variables, the distribution law of multiplication is written as follows:

(a + b) × c = a × c + b × c

where instead of a, b, c Any numbers can be.

Law of multiplication by zero

This law says that if there is at least one zero in any multiplication, then the answer will be zero.

The product is equal to zero if at least one of the factors is equal to zero.

For example, the expression 0 × 2 is equal to zero

In this case, the number 2 is a multiplier and shows how many times the multiplicand needs to be increased. That is, how many times to increase zero. Literally this expression reads like this: "double zero" . But how can you double a zero if it is zero? The answer is no.

In other words, if “nothing” is doubled or even a million times, it will still turn out to be “nothing.”

And if you swap the factors in the expression 0 × 2, you will again get zero. We know this from the previous displacement law:

Examples of applying the law of multiplication by zero:

5 × 5 × 5 × 0 = 0

2 × 5 × 0 × 9 × 1 = 0

In the last two examples there are several factors. Having seen a zero in them, we immediately put a zero in the answer, applying the law of multiplication by zero.

We looked at the basic laws of multiplication. Next, we'll look at multiplying integers.

Multiplying Integers

Example 1. Find the value of the expression −5 × 2

This is the multiplication of numbers with different signs. −5 is a negative number and 2 is a positive number. For such cases, the following rule should be applied:

To multiply numbers with different signs, you need to multiply their modules and put a minus in front of the resulting answer.

−5 × 2 = − (|−5| × |2|) = − (5 × 2) = − (10) = −10

Usually written shorter: −5 × 2 = −10

Any multiplication can be represented as a sum of numbers. For example, consider the expression 2 × 3. It equals 6.

The multiplier in this expression is the number 3. This multiplier shows how many times you need to increase the two. But the expression 2 × 3 can also be understood as the sum of three twos:

The same thing happens with the expression −5 × 2. This expression can be represented as the sum

And the expression (−5) + (−5) is equal to −10. We know this from . This is the addition of negative numbers. Recall that the result of adding negative numbers is a negative number.

Example 2. Find the value of the expression 12 × (−5)

This is the multiplication of numbers with different signs. 12 is a positive number, (−5) is negative. Again we apply the previous rule. We multiply the modules of numbers and put a minus in front of the resulting answer:

12 × (−5) = − (|12| × |−5|) = − (12 × 5) = − (60) = −60

Usually the solution is written shorter:

12 × (−5) = −60

Example 3. Find the value of the expression 10 × (−4) × 2

This expression consists of several factors. First, multiply 10 and (−4), then multiply the resulting number by 2. Along the way, apply the previously learned rules:

First action:

10 × (−4) = −(|10| × |−4|) = −(10 × 4) = (−40) = −40

Second action:

−40 × 2 = −(|−40 | × | 2|) = −(40 × 2) = −(80) = −80

So the value of the expression 10 × (−4) × 2 is −80

Let's write down the solution briefly:

10 × (−4) × 2 = −40 × 2 = −80

Example 4. Find the value of the expression (−4) × (−2)

This is the multiplication of negative numbers. In such cases, the following rule must be applied:

To multiply negative numbers, you need to multiply their modules and put a plus in front of the resulting answer.

(−4) × (−2) = |−4| × |−2| = 4 × 2 = 8

Traditionally, we don’t write down the plus, so we just write down the answer 8.

Let's write the solution shorter (−4) × (−2) = 8

The question arises: why does multiplying negative numbers suddenly produce a positive number? Let's try to prove that (−4) × (−2) equals 8 and nothing else.

First we write the following expression:

Let's enclose it in brackets:

(4 × (−2))

Let's add to this expression our expression (−4) × (−2). Let's put it in brackets too:

(4 × (−2) ) + ((−4) × (−2) )

Let's equate all this to zero:

(4 × (−2)) + ((−4) × (−2)) = 0

Now the fun begins. The point is that we must evaluate the left side of this expression and get 0 as a result.

So the first product (4 × (−2)) is −8. Let's write the number −8 in our expression instead of the product (4 × (−2))

−8 + ((−4) × (−2)) = 0

Now instead of the second work we will temporarily put an ellipsis

Now let's look carefully at the expression −8 + ... = 0. What number should stand in place of the ellipsis for equality to be maintained? The answer suggests itself. Instead of an ellipsis there should be a positive number 8 and nothing else. This is the only way equality will be maintained. After all, −8 + 8 equals 0.

We return to the expression −8 + ((−4) × (−2)) = 0 and instead of the product ((−4) × (−2)) we write the number 8

Example 5. Find the value of the expression −2 × (6 + 4)

Let's apply the distributive law of multiplication, that is, multiply the number −2 by each term of the sum (6 + 4)

−2 × (6 + 4) = −2 × 6 + (−2) × 4

Now let's do the multiplication and add up the results. Along the way, we apply the previously learned rules. The entry with modules can be skipped so as not to clutter the expression

First action:

−2 × 6 = −12

Second action:

−2 × 4 = −8

Third action:

−12 + (−8) = −20

So the value of the expression −2 × (6 + 4) is −20

Let's write down the solution briefly:

−2 × (6 + 4) = (−12) + (−8) = −20

Example 6. Find the value of the expression (−2) × (−3) × (−4)

The expression consists of several factors. First, multiply the numbers −2 and −3, and multiply the resulting product by the remaining number −4. Let's skip the entry with modules so as not to clutter the expression

First action:

(−2) × (−3) = 6

Second action:

6 × (−4) = −(6 × 4) = −24

So the value of the expression (−2) × (−3) × (−4) is equal to −24

Let's write down the solution briefly:

(−2) × (−3) × (−4) = 6 × (−4) = −24

Laws of division

Before dividing integers, you need to learn the two laws of division.

First of all, let’s remember what division consists of. The division consists of three parameters: divisible, divisor And private. For example, in expression 8: 2 = 4, 8 is the dividend, 2 is the divisor, 4 is the quotient.

Dividend shows what exactly we are sharing. In our example we are dividing the number 8.

Divider shows how many parts the dividend must be divided into. In our example, the divisor is the number 2. This divisor shows how many parts the dividend 8 needs to be divided into. That is, during the division operation, the number 8 will be divided into two parts.

Private- This is the actual result of the division operation. In our example, the quotient is 4. This quotient is the result of dividing 8 by 2.

You can't divide by zero

Any number cannot be divided by zero.

The fact is that division is the inverse action of multiplication. This phrase can be understood in its literal sense. For example, if 2 × 5 = 10, then 10:5 = 2.

It can be seen that the second expression is written in reverse order. If, for example, we have two apples and we want to increase them five times, then we will write 2 × 5 = 10. The result will be ten apples. Then, if we want to reduce those ten apples back down to two, we write 10: 5 = 2

You can do the same with other expressions. If, for example, 2 × 6 = 12, then we can return back to the original number 2. To do this, just write the expression 2 × 6 = 12 in reverse order, dividing 12 by 6

Now consider the expression 5 × 0. We know from the laws of multiplication that the product is equal to zero if at least one of the factors is equal to zero. This means that the expression 5 × 0 is equal to zero

If we write this expression in reverse order, we get:

The answer that immediately catches your eye is 5, which is obtained by dividing zero by zero. This is impossible.

In reverse order, you can write another similar expression, for example 2 × 0 = 0

In the first case, dividing zero by zero we got 5, and in the second case 2. That is, each time dividing zero by zero, we can get different values, and this is unacceptable.

The second explanation is that dividing the dividend by the divisor means finding a number that, when multiplied by the divisor, gives the dividend.

For example, the expression 8: 2 means finding a number that, when multiplied by 2, gives 8

Here, instead of an ellipsis, there should be a number that, when multiplied by 2, will give the answer 8. To find this number, just write this expression in reverse order:

We got the number 4. Let's write it instead of the ellipsis:

Now imagine that you need to find the value of the expression 5: 0. In this case, 5 is the dividend, 0 is the divisor. Dividing 5 by 0 means finding a number that when multiplied by 0 gives 5

Here, instead of an ellipsis, there should be a number that, when multiplied by 0, will give the answer 5. But there is no number that, when multiplied by zero, gives 5.

The expression ... × 0 = 5 contradicts the law of multiplication by zero, which states that the product is equal to zero when at least one of the factors is equal to zero.

This means that writing the expression... × 0 = 5 in reverse order, dividing 5 by 0 makes no sense. That's why they say you can't divide by zero.

Using variables, this law is written as follows:

At b ≠ 0

Number a can be divided by a number b, provided that b not equal to zero.

Property of private

This law says that if the dividend and the divisor are multiplied or divided by the same number, the quotient will not change.

For example, consider expression 12: 4. The value of this expression is 3

Let's try to multiply the dividend and divisor by the same number, for example by the number 4. If we believe the property of the quotient, we should again get the number 3 in the answer

(12 × 4) : (4 × 4)

(12 × 4) : (4 × 4) = 48: 16 = 3

We received answer 3.

Now let's try not to multiply, but to divide the dividend and divisor by the number 4

(12: 4 ) : (4: 4 )

(12: 4 ) : (4: 4 ) = 3: 1 = 3

We received answer 3.

We see that if the dividend and divisor are multiplied or divided by the same number, then the quotient does not change.

Integer division

Example 1. Find the value of expression 12: (−2)

This is the division of numbers with different signs. 12 is a positive number, (−2) is negative. To solve this example, you need Divide the module of the dividend by the module of the divisor, and put a minus before the resulting answer.

12: (−2) = −(|12| : |−2|) = −(12: 2) = −(6) = −6

Usually written shorter:

12: (−2) = −6

Example 2. Find the value of the expression −24: 6

This is the division of numbers with different signs. −24 is a negative number, 6 is a positive number. Yet again Divide the module of the dividend by the module of the divisor, and put a minus in front of the resulting answer.

−24: 6 = −(|−24| : |6|) = −(24: 6) = −(4) = −4

Let's write down the solution briefly:

Example 3. Find the value of the expression −45: (−5)

This is division of negative numbers. To solve this example, you need Divide the module of the dividend by the module of the divisor, and put a plus sign in front of the resulting answer.

−45: (−5) = |−45| : |−5| = 45: 5 = 9

Let's write down the solution briefly:

−45: (−5) = 9

Example 4. Find the value of the expression −36: (−4) : (−3)

According to, if the expression contains only multiplication or division, then all actions must be performed from left to right in the order they appear.

Divide −36 by (−4), and divide the resulting number by −3

First action:

−36: (−4) = |−36| : |−4| = 36: 4 = 9

Second action:

9: (−3) = −(|9| : |−3|) = −(9: 3) = −(3) = −3

Let's write down the solution briefly:

−36: (−4) : (−3) = 9: (−3) = −3

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Primary school is coming to an end, and soon the child will step into the advanced world of mathematics. But already during this period the student is faced with the difficulties of science. When performing a simple task, the child gets confused and lost, which ultimately leads to a negative mark for the work done. To avoid such troubles, when solving examples, you need to be able to navigate in the order in which you need to solve the example. Having distributed the actions incorrectly, the child does not complete the task correctly. The article reveals the basic rules for solving examples that contain the entire range of mathematical calculations, including brackets. Procedure in mathematics 4th grade rules and examples.

Before completing the task, ask your child to number the actions that he is going to perform. If you have any difficulties, please help.

Some rules to follow when solving examples without brackets:

If a task requires a number of actions to be performed, you must first perform division or multiplication, then . All actions are performed as the letter progresses. Otherwise, the result of the decision will not be correct.

If in the example you need to execute, we do it in order, from left to right.

27-5+15=37 (When solving the example, we are guided by the rule. First we perform subtraction, then addition).

Teach your child to always plan and number the actions performed.

The answers to each solved action are written above the example. This will make it much easier for the child to navigate the actions.

Let's consider another option where it is necessary to distribute actions in order:

As you can see, when solving, the rule is followed: first we look for the product, then we look for the difference.

These are simple examples that require careful consideration when solving them. Many children are stunned when they see a task that contains not only multiplication and division, but also parentheses. A student who does not know the procedure for performing actions has questions that prevent him from completing the task.

As stated in the rule, first we find the product or quotient, and then everything else. But there are parentheses! What to do in this case?

Solving examples with brackets

Let's look at a specific example:

When performing this task, we first find the value of the expression enclosed in parentheses.
You should start with multiplication, then addition.
After the expression in brackets is solved, we proceed to actions outside them.
According to the rules of procedure, the next step is multiplication.
The final stage will be.

As we can see in the visual example, all actions are numbered. To reinforce the topic, invite your child to solve several examples on their own:

The order in which the value of the expression should be calculated has already been arranged. The child will only have to carry out the decision directly.

Let's complicate the task. Let the child find the meaning of the expressions on his own.

7*3-5*4+(20-19) 14+2*3-(13-9)
17+2*5+(28-2) 5*3+15-(2-1*2)
24-3*2-(56-4*3) 14+12-3*(21-7)

Teach your child to solve all tasks in draft form. In this case, the student will have the opportunity to correct an incorrect decision or blots. Corrections are not allowed in the workbook. By completing tasks on their own, children see their mistakes.

Parents, in turn, should pay attention to mistakes, help the child understand and correct them. You shouldn’t overload a student’s brain with large amounts of tasks. With such actions you will discourage the child’s desire for knowledge. There should be a sense of proportion in everything.

Take a break. The child should be distracted and take a break from classes. The main thing to remember is that not everyone has a mathematical mind. Maybe your child will grow up to be a famous philosopher.

This lesson discusses in detail the procedure for performing arithmetic operations in expressions without parentheses and with brackets. Students are given the opportunity, while completing assignments, to determine whether the meaning of expressions depends on the order in which arithmetic operations are performed, to find out whether the order of arithmetic operations is different in expressions without parentheses and with parentheses, to practice applying the learned rule, to find and correct errors made when determining the order of actions.

In life, we constantly perform some kind of action: we walk, study, read, write, count, smile, quarrel and make peace. We perform these actions in different orders. Sometimes they can be swapped, sometimes not. For example, when getting ready for school in the morning, you can first do exercises, then make your bed, or vice versa. But you can’t go to school first and then put on clothes.

In mathematics, is it necessary to perform arithmetic operations in a certain order?

Let's check

Let's compare the expressions:
8-3+4 and 8-3+4

We see that both expressions are exactly the same.

Let's perform actions in one expression from left to right, and in the other from right to left. You can use numbers to indicate the order of actions (Fig. 1).

Rice. 1. Procedure

In the first expression, we will first perform the subtraction operation and then add the number 4 to the result.

In the second expression, we first find the value of the sum, and then subtract the resulting result 7 from 8.

We see that the meanings of the expressions are different.

Let's conclude: The order in which arithmetic operations are performed cannot be changed.

Let's learn the rule for performing arithmetic operations in expressions without parentheses.

If an expression without parentheses includes only addition and subtraction or only multiplication and division, then the actions are performed in the order in which they are written.

Let's practice.

Consider the expression

This expression contains only addition and subtraction operations. These actions are called first stage actions.

We perform the actions from left to right in order (Fig. 2).

Rice. 2. Procedure

Consider the second expression

This expression contains only multiplication and division operations - These are the actions of the second stage.

We perform the actions from left to right in order (Fig. 3).

Rice. 3. Procedure

In what order are arithmetic operations performed if the expression contains not only addition and subtraction, but also multiplication and division?

If an expression without parentheses includes not only the operations of addition and subtraction, but also multiplication and division, or both of these operations, then first perform in order (from left to right) multiplication and division, and then addition and subtraction.

Let's look at the expression.

Let's think like this. This expression contains the operations of addition and subtraction, multiplication and division. We act according to the rule. First, we perform in order (from left to right) multiplication and division, and then addition and subtraction. Let's arrange the order of actions.

Let's calculate the value of the expression.

18:2-2*3+12:3=9-6+4=3+4=7

In what order are arithmetic operations performed if there are parentheses in an expression?

If an expression contains parentheses, the value of the expressions in the parentheses is evaluated first.

Let's look at the expression.

30 + 6 * (13 - 9)

We see that in this expression there is an action in parentheses, which means we will perform this action first, then multiplication and addition in order. Let's arrange the order of actions.

30 + 6 * (13 - 9)

Let's calculate the value of the expression.

30+6*(13-9)=30+6*4=30+24=54

How should one reason to correctly establish the order of arithmetic operations in a numerical expression?

Before starting calculations, you need to look at the expression (find out whether it contains parentheses, what actions it contains) and only then perform the actions in the following order:

1. actions written in brackets;

2. multiplication and division;

3. addition and subtraction.

The diagram will help you remember this simple rule (Fig. 4).

Rice. 4. Procedure

Let's practice.

Let's consider the expressions, establish the order of actions and perform calculations.

43 - (20 - 7) +15

32 + 9 * (19 - 16)

We will act according to the rule. The expression 43 - (20 - 7) +15 contains operations in parentheses, as well as addition and subtraction operations. Let's establish a procedure. The first action is to perform the operation in parentheses, and then, in order from left to right, subtraction and addition.

43 - (20 - 7) +15 =43 - 13 +15 = 30 + 15 = 45

The expression 32 + 9 * (19 - 16) contains operations in parentheses, as well as multiplication and addition operations. According to the rule, we first perform the action in parentheses, then multiplication (we multiply the number 9 by the result obtained by subtraction) and addition.

32 + 9 * (19 - 16) =32 + 9 * 3 = 32 + 27 = 59

In the expression 2*9-18:3 there are no parentheses, but there are multiplication, division and subtraction operations. We act according to the rule. First, we perform multiplication and division from left to right, and then subtract the result obtained from division from the result obtained by multiplication. That is, the first action is multiplication, the second is division, and the third is subtraction.

2*9-18:3=18-6=12

Let's find out whether the order of actions in the following expressions is correctly defined.

37 + 9 - 6: 2 * 3 =

18: (11 - 5) + 47=

7 * 3 - (16 + 4)=

Let's think like this.

37 + 9 - 6: 2 * 3 =

There are no parentheses in this expression, which means that we first perform multiplication or division from left to right, then addition or subtraction. In this expression, the first action is division, the second is multiplication. The third action should be addition, the fourth - subtraction. Conclusion: the procedure is determined correctly.

Let's find the value of this expression.

37+9-6:2*3 =37+9-3*3=37+9-9=46-9=37

Let's continue to talk.

The second expression contains parentheses, which means that we first perform the action in parentheses, then from left to right multiplication or division, addition or subtraction. We check: the first action is in parentheses, the second is division, the third is addition. Conclusion: the procedure is defined incorrectly. Let's correct the errors and find the meaning of the expression.

18:(11-5)+47=18:6+47=3+47=50

This expression also contains parentheses, which means that we first perform the action in parentheses, then from left to right multiplication or division, addition or subtraction. Let's check: the first action is in parentheses, the second is multiplication, the third is subtraction. Conclusion: the procedure is defined incorrectly. Let's correct the errors and find the meaning of the expression.

7*3-(16+4)=7*3-20=21-20=1

Let's complete the task.

Let's arrange the order of actions in the expression using the learned rule (Fig. 5).

Rice. 5. Procedure

We don't see numerical values, so we won't be able to find the meaning of expressions, but we'll practice applying the rule we've learned.

We act according to the algorithm.

The first expression contains parentheses, which means the first action is in parentheses. Then from left to right multiplication and division, then from left to right subtraction and addition.

The second expression also contains parentheses, which means we perform the first action in parentheses. After that, from left to right, multiplication and division, after that, subtraction.

Let's check ourselves (Fig. 6).

Rice. 6. Procedure

Today in class we learned about the rule for the order of actions in expressions without and with brackets.

Bibliography

M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 1. - M.: “Enlightenment”, 2012.
M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 2. - M.: “Enlightenment”, 2012.
M.I. Moro. Mathematics lessons: Methodological recommendations for teachers. 3rd grade. - M.: Education, 2012.
Regulatory document. Monitoring and evaluation of learning outcomes. - M.: “Enlightenment”, 2011.
“School of Russia”: Programs for primary school. - M.: “Enlightenment”, 2011.
S.I. Volkova. Mathematics: Test papers. 3rd grade. - M.: Education, 2012.
V.N. Rudnitskaya. Tests. - M.: “Exam”, 2012.

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Openclass.ru ().

Homework

1. Determine the order of actions in these expressions. Find the meaning of the expressions.

2. Determine in what expression this order of actions is performed:

1. multiplication; 2. division;. 3. addition; 4. subtraction; 5. addition. Find the meaning of this expression.

3. Make up three expressions in which the following order of actions is performed:

1. multiplication; 2. addition; 3. subtraction

1. addition; 2. subtraction; 3. addition

1. multiplication; 2. division; 3. addition

Find the meaning of these expressions.

Division is one of the four basic mathematical operations (addition, subtraction, multiplication). Division, like other operations, is important not only in mathematics, but also in everyday life. For example, you as a whole class (25 people) donate money and buy a gift for the teacher, but you don’t spend it all, there will be change left over. So you will need to divide the change among everyone. The division operation comes into play to help you solve this problem.

Division is an interesting operation, as we will see in this article!

Dividing numbers

So, a little theory, and then practice! What is division? Division is breaking something into equal parts. That is, it could be a bag of sweets that needs to be divided into equal parts. For example, there are 9 candies in a bag, and the person who wants to receive them is three. Then you need to divide these 9 candies among three people.

It is written like this: 9:3, the answer will be the number 3. That is, dividing the number 9 by the number 3 shows the number of three numbers contained in the number 9. The reverse action, a check, will be multiplication. 3*3=9. Right? Absolutely.

So let's look at example 12:6. First, let's name each component of the example. 12 – dividend, that is. a number that can be divided into parts. 6 is a divisor, this is the number of parts into which the dividend is divided. And the result will be a number called “quotient”.

Let's divide 12 by 6, the answer will be the number 2. You can check the solution by multiplying: 2*6=12. It turns out that the number 6 is contained 2 times in the number 12.

Division with remainder

What is division with a remainder? This is the same division, only the result is not an even number, as shown above.

For example, let's divide 17 by 5. Since the largest number divisible by 5 to 17 is 15, then the answer will be 3 and the remainder is 2, and is written like this: 17:5 = 3(2).

For example, 22:7. In the same way, we determine the maximum number divisible by 7 to 22. This number is 21. The answer then will be: 3 and the remainder 1. And it is written: 22:7 = 3 (1).

Division by 3 and 9

A special case of division would be division by the number 3 and the number 9. If you want to find out whether a number is divisible by 3 or 9 without a remainder, then you will need:

Find the sum of the digits of the dividend.

Divide by 3 or 9 (depending on what you need).

If the answer is obtained without a remainder, then the number will be divided without a remainder.

For example, the number 18. The sum of the digits is 1+8 = 9. The sum of the digits is divisible by both 3 and 9. The number 18:9=2, 18:3=6. Divided without remainder.

For example, the number 63. The sum of the digits is 6+3 = 9. Divisible by both 9 and 3. 63:9 = 7, and 63:3 = 21. Such operations are carried out with any number to find out whether it is divisible with the remainder by 3 or 9, or not.

Multiplication and division

Multiplication and division are opposite operations. Multiplication can be used as a test for division, and division can be used as a test for multiplication. You can learn more about multiplication and master the operation in our article about multiplication. Which describes multiplication in detail and how to do it correctly. There you will also find the multiplication table and examples for training.

Here is an example of checking division and multiplication. Let's say the example is 6*4. Answer: 24. Then let's check the answer by division: 24:4=6, 24:6=4. It was decided correctly. In this case, the check is performed by dividing the answer by one of the factors.

Or an example is given for the division 56:8. Answer: 7. Then the test will be 8*7=56. Right? Yes. In this case, the test is performed by multiplying the answer by the divisor.

Division 3 class

In third grade they are just starting to go through division. Therefore, third graders solve the simplest problems:

Problem 1. A factory worker was given the task of putting 56 cakes into 8 packages. How many cakes should be put in each package to make the same amount in each?

Problem 2. On New Year's Eve at school, children in a class of 15 students were given 75 candies. How many candies should each child receive?

Problem 3. Roma, Sasha and Misha picked 27 apples from the apple tree. How many apples will each person get if they need to be divided equally?

Problem 4. Four friends bought 58 cookies. But then they realized that they could not divide them equally. How many additional cookies do the kids need to buy so that each gets 15?

Division 4th grade

The division in the fourth grade is more serious than in the third. All calculations are carried out using the column division method, and the numbers involved in the division are not small. What is long division? You can find the answer below:

Column division

What is long division? This is a method that allows you to find the answer to dividing large numbers. If prime numbers like 16 and 4 can be divided, and the answer is clear - 4. Then 512:8 is not easy for a child in his mind. And it’s our task to talk about the technique for solving such examples.

Let's look at an example, 512:8.

1 step. Let's write the dividend and divisor as follows:

The quotient will ultimately be written under the divisor, and the calculations under the dividend.

Step 2. We start dividing from left to right. First we take the number 5:

Step 3. The number 5 is less than the number 8, which means it will not be possible to divide. Therefore, we take another digit of the dividend:

Now 51 is greater than 8. This is an incomplete quotient.

Step 4. We put a dot under the divisor.

Step 5. After 51 there is another number 2, which means there will be one more number in the answer, that is. quotient is a two-digit number. Let's put the second point:

Step 6. We begin the division operation. The largest number divisible by 8 without a remainder to 51 is 48. Dividing 48 by 8, we get 6. Write the number 6 instead of the first dot under the divisor:

Step 7. Then write the number exactly below the number 51 and put a “-” sign:

Step 8. Then we subtract 48 from 51 and get the answer 3.

* 9 step*. We take down the number 2 and write it next to the number 3:

Step 10 We divide the resulting number 32 by 8 and get the second digit of the answer – 4.

So the answer is 64, without remainder. If we divided the number 513, then the remainder would be one.

Division of three digits

Dividing three-digit numbers is done using the long division method, which was explained in the example above. An example of just a three-digit number.

Division of fractions

Dividing fractions is not as difficult as it seems at first glance. For example, (2/3):(1/4). The method of this division is quite simple. 2/3 is the dividend, 1/4 is the divisor. You can replace the division sign (:) with multiplication ( ), but to do this you need to swap the numerator and denominator of the divisor. That is, we get: (2/3)(4/1), (2/3)*4, this is equal to 8/3 or 2 integers and 2/3. Let's give another example, with an illustration for better understanding. Consider the fractions (4/7):(2/5):

As in the previous example, we reverse the 2/5 divisor and get 5/2, replacing division with multiplication. We then get (4/7)*(5/2). We make a reduction and answer: 10/7, then take out the whole part: 1 whole and 3/7.

Dividing numbers into classes

Let's imagine the number 148951784296, and divide it into three digits: 148,951,784,296. So, from right to left: 296 is the class of units, 784 is the class of thousands, 951 is the class of millions, 148 is the class of billions. In turn, in each class 3 digits have their own digit. From right to left: the first digit is units, the second digit is tens, the third is hundreds. For example, the class of units is 296, 6 is ones, 9 is tens, 2 is hundreds.

Division of natural numbers

Division of natural numbers is the simplest division described in this article. It can be either with or without a remainder. The divisor and dividend can be any non-fractional, integer numbers.

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Division presentation

Presentation is another way to visualize the topic of division. Below we will find a link to an excellent presentation that does a good job of explaining how to divide, what division is, what dividend, divisor and quotient are. Don’t waste your time, but consolidate your knowledge!

Examples for division

Easy level

Average level

Difficult level

Games for developing mental arithmetic

Special educational games developed with the participation of Russian scientists from Skolkovo will help improve mental arithmetic skills in an interesting game form.

Game "Guess the operation"

The game “Guess the Operation” develops thinking and memory. The main point of the game is to choose a mathematical sign for the equality to be true. Examples are given on the screen, look carefully and put the required “+” or “-” sign so that the equality is true. The “+” and “-” signs are located at the bottom of the picture, select the desired sign and click on the desired button. If you answered correctly, you score points and continue playing.

Game "Simplification"

The game “Simplification” develops thinking and memory. The main essence of the game is to quickly perform a mathematical operation. A student is drawn on the screen at the blackboard, and a mathematical operation is given; the student needs to calculate this example and write the answer. Below are three answers, count and click the number you need using the mouse. If you answered correctly, you score points and continue playing.

Game "Quick addition"

The game "Quick Addition" develops thinking and memory. The main essence of the game is to choose numbers whose sum is equal to a given number. In this game, a matrix from one to sixteen is given. A given number is written above the matrix; you need to select the numbers in the matrix so that the sum of these digits is equal to the given number. If you answered correctly, you score points and continue playing.

Visual Geometry Game

The game "Visual Geometry" develops thinking and memory. The main essence of the game is to quickly count the number of shaded objects and select it from the list of answers. In this game, blue squares are shown on the screen for a few seconds, you need to quickly count them, then they close. Below the table there are four numbers written, you need to select one correct number and click on it with the mouse. If you answered correctly, you score points and continue playing.

Game "Piggy Bank"

The Piggy Bank game develops thinking and memory. The main essence of the game is to choose which piggy bank has more money. In this game there are four piggy banks, you need to count which piggy bank has the most money and show this piggy bank with the mouse. If you answered correctly, then you score points and continue playing.

Game "Fast addition reload"

The game “Fast addition reboot” develops thinking, memory and attention. The main point of the game is to choose the correct terms, the sum of which will be equal to the given number. In this game, three numbers are given on the screen and a task is given, add the number, the screen indicates which number needs to be added. You select the desired numbers from three numbers and press them. If you answered correctly, then you score points and continue playing.

Development of phenomenal mental arithmetic

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In mathematics, is it necessary to perform arithmetic operations in a certain order?

Let's check

Let's compare the expressions:
8-3+4 and 8-3+4

We see that both expressions are exactly the same.

Let's perform actions in one expression from left to right, and in the other from right to left. You can use numbers to indicate the order of actions (Fig. 1).

Rice. 1. Procedure

In the first expression, we will first perform the subtraction operation and then add the number 4 to the result.

In the second expression, we first find the value of the sum, and then subtract the resulting result 7 from 8.

We see that the meanings of the expressions are different.

Let's conclude: The order in which arithmetic operations are performed cannot be changed.

Let's learn the rule for performing arithmetic operations in expressions without parentheses.

If an expression without parentheses includes only addition and subtraction or only multiplication and division, then the actions are performed in the order in which they are written.

Let's practice.

Consider the expression

This expression contains only addition and subtraction operations. These actions are called first stage actions.

We perform the actions from left to right in order (Fig. 2).

Rice. 2. Procedure

Consider the second expression

This expression contains only multiplication and division operations - These are the actions of the second stage.

We perform the actions from left to right in order (Fig. 3).

Rice. 3. Procedure

In what order are arithmetic operations performed if the expression contains not only addition and subtraction, but also multiplication and division?

Let's look at the expression.

Let's calculate the value of the expression.

18:2-2*3+12:3=9-6+4=3+4=7

In what order are arithmetic operations performed if there are parentheses in an expression?

If an expression contains parentheses, the value of the expressions in the parentheses is evaluated first.

Let's look at the expression.

30 + 6 * (13 - 9)

We see that in this expression there is an action in parentheses, which means we will perform this action first, then multiplication and addition in order. Let's arrange the order of actions.

30 + 6 * (13 - 9)

Let's calculate the value of the expression.

30+6*(13-9)=30+6*4=30+24=54

How should one reason to correctly establish the order of arithmetic operations in a numerical expression?

Before starting calculations, you need to look at the expression (find out whether it contains parentheses, what actions it contains) and only then perform the actions in the following order:

1. actions written in brackets;

2. multiplication and division;

3. addition and subtraction.

The diagram will help you remember this simple rule (Fig. 4).

Rice. 4. Procedure

Let's practice.

Let's consider the expressions, establish the order of actions and perform calculations.

43 - (20 - 7) +15

32 + 9 * (19 - 16)

43 - (20 - 7) +15 =43 - 13 +15 = 30 + 15 = 45

32 + 9 * (19 - 16) =32 + 9 * 3 = 32 + 27 = 59

2*9-18:3=18-6=12

Let's find out whether the order of actions in the following expressions is correctly defined.

37 + 9 - 6: 2 * 3 =

18: (11 - 5) + 47=

7 * 3 - (16 + 4)=

Let's think like this.

37 + 9 - 6: 2 * 3 =

Let's find the value of this expression.

37+9-6:2*3 =37+9-3*3=37+9-9=46-9=37

Let's continue to talk.

18:(11-5)+47=18:6+47=3+47=50

7*3-(16+4)=7*3-20=21-20=1

Let's complete the task.

Let's arrange the order of actions in the expression using the learned rule (Fig. 5).

Rice. 5. Procedure

We don't see numerical values, so we won't be able to find the meaning of expressions, but we'll practice applying the rule we've learned.

We act according to the algorithm.

The first expression contains parentheses, which means the first action is in parentheses. Then from left to right multiplication and division, then from left to right subtraction and addition.

The second expression also contains parentheses, which means we perform the first action in parentheses. After that, from left to right, multiplication and division, after that, subtraction.

Let's check ourselves (Fig. 6).

Rice. 6. Procedure

Today in class we learned about the rule for the order of actions in expressions without and with brackets.

Bibliography

M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 1. - M.: “Enlightenment”, 2012.
M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 2. - M.: “Enlightenment”, 2012.
M.I. Moro. Mathematics lessons: Methodological recommendations for teachers. 3rd grade. - M.: Education, 2012.
Regulatory document. Monitoring and evaluation of learning outcomes. - M.: “Enlightenment”, 2011.
“School of Russia”: Programs for primary school. - M.: “Enlightenment”, 2011.
S.I. Volkova. Mathematics: Test papers. 3rd grade. - M.: Education, 2012.
V.N. Rudnitskaya. Tests. - M.: “Exam”, 2012.

Festival.1september.ru ().
Sosnovoborsk-soobchestva.ru ().
Openclass.ru ().

Homework

1. Determine the order of actions in these expressions. Find the meaning of the expressions.

2. Determine in what expression this order of actions is performed:

1. multiplication; 2. division;. 3. addition; 4. subtraction; 5. addition. Find the meaning of this expression.

3. Make up three expressions in which the following order of actions is performed:

1. multiplication; 2. addition; 3. subtraction

1. addition; 2. subtraction; 3. addition

1. multiplication; 2. division; 3. addition

Find the meaning of these expressions.