According to the rule for performing arithmetic. Learning the rules of procedure

When calculating examples, you must observe a certain order actions. Using the rules below, we will figure out the order in which the actions are performed and what the parentheses are for.

If there are no parentheses in the expression, then:

first we perform all the operations of multiplication and division from left to right;

and then from left to right all the addition and subtraction operations.

Let's consider procedure in the following example.

We remind you that order of operations in mathematics arranged from left to right (from the beginning to the end of the example).

When calculating the value of an expression, you can record it in two ways.

First way

Each action is recorded separately with its own number under the example.
After the last action is completed, the response is necessarily written to the original example.

When calculating the results of actions with two-digit and/or three-digit numbers Be sure to list your calculations in a column.

Second way

The second method is called chain recording. All calculations are carried out in exactly the same order, but the results are written immediately after the equal sign.

If the expression contains parentheses, then the actions in the parentheses are performed first.

Inside the parentheses themselves, the order of actions is the same as in expressions without parentheses.

If there are more brackets inside the brackets, then the actions inside the nested (inner) brackets are performed first.

Procedure and exponentiation

If the example contains a numeric or literal expression in brackets that must be raised to a power, then:

First we perform all the actions inside the brackets
Then we raise to a power all parentheses and numbers that stand in a power, from left to right (from the beginning to the end of the example).
We carry out the remaining steps as usual

Procedure for performing actions, rules, examples.

Numeric, literal expressions and expressions with variables in their notation may contain signs of various arithmetic operations. When transforming expressions and calculating the values of expressions, actions are performed in a certain order, in other words, you must observe order of actions.

In this article, we will figure out which actions should be performed first and which ones after them. Let's start with the simplest cases, when the expression contains only numbers or variables connected by plus, minus, multiply and divide signs. Next, we will explain what order of actions should be followed in expressions with brackets. Finally, let's look at the order in which actions are performed in expressions containing powers, roots, and other functions.

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First multiplication and division, then addition and subtraction

The school gives the following a rule that determines the order in which actions are performed in expressions without parentheses:

actions are performed in order from left to right,
Moreover, multiplication and division are performed first, and then addition and subtraction.

The stated rule is perceived quite naturally. Performing actions in order from left to right is explained by the fact that we usually keep records from left to right. And the fact that multiplication and division are performed before addition and subtraction is explained by the meaning that these actions carry.

Let's look at a few examples of how this rule applies. For examples we will take the simplest numeric expressions, so as not to be distracted by calculations, but to focus specifically on the order of actions.

Follow steps 7−3+6.

The original expression does not contain parentheses, and it does not contain multiplication or division. Therefore, we should perform all the actions in order from left to right, that is, first we subtract 3 from 7, we get 4, after which we add 6 to the resulting difference of 4, we get 10.

Briefly, the solution can be written as follows: 7−3+6=4+6=10.

Indicate the order of actions in the expression 6:2·8:3.

To answer the question of the problem, let's turn to the rule indicating the order of execution of actions in expressions without parentheses. The original expression contains only multiplication and division operations, and according to the rule, they must be performed in order from left to right.

First we divide 6 by 2, multiply this quotient by 8, and finally divide the result by 3.

Calculate the value of the expression 17−5·6:3−2+4:2.

First, let's determine in what order the actions in the original expression should be performed. It contains both multiplication and division and addition and subtraction. First, from left to right, you need to perform multiplication and division. So we multiply 5 by 6, we get 30, we divide this number by 3, we get 10. Now we divide 4 by 2, we get 2. We substitute the found value 10 into the original expression instead of 5·6:3, and instead of 4:2 - the value 2, we have 17−5·6:3−2+4:2=17−10−2+2.

The resulting expression no longer contains multiplication and division, so it remains to perform the remaining actions in order from left to right: 17−10−2+2=7−2+2=5+2=7.

At first, in order not to confuse the order in which actions are performed when calculating the value of an expression, it is convenient to place numbers above the action signs that correspond to the order in which they are performed. For the previous example it would look like this: .

The same order of operations - first multiplication and division, then addition and subtraction - should be followed when working with letter expressions.

Actions of the first and second stages

In some mathematics textbooks there is a division of arithmetic operations into operations of the first and second stages. Let's figure this out.

Actions of the first stage addition and subtraction are called, and multiplication and division are called second stage actions.

In these terms, the rule from the previous paragraph, which determines the order of execution of actions, will be written as follows: if the expression does not contain parentheses, then in order from left to right, the actions of the second stage (multiplication and division) are performed first, then the actions of the first stage (addition and subtraction).

Order of arithmetic operations in expressions with parentheses

Expressions often contain parentheses to indicate the order in which actions are performed. In this case a rule that specifies the order of execution of actions in expressions with parentheses, is formulated as follows: first, the actions in brackets are performed, while multiplication and division are also performed in order from left to right, then addition and subtraction.

So, the expressions in brackets are considered as components of the original expression, and they retain the order of actions already known to us. Let's look at the solutions to the examples for greater clarity.

Follow these steps 5+(7−2·3)·(6−4):2.

The expression contains parentheses, so let's first perform the actions in the expressions enclosed in these parentheses. Let's start with the expression 7−2·3. In it you must first perform multiplication, and only then subtraction, we have 7−2·3=7−6=1. Let's move on to the second expression in brackets 6−4. There is only one action here - subtraction, we perform it 6−4 = 2.

We substitute the obtained values into the original expression: 5+(7−2·3)·(6−4):2=5+1·2:2. In the resulting expression, we first perform multiplication and division from left to right, then subtraction, we get 5+1·2:2=5+2:2=5+1=6. At this point, all actions are completed, we adhered to the following order of their implementation: 5+(7−2·3)·(6−4):2.

Let's write down a short solution: 5+(7−2·3)·(6−4):2=5+1·2:2=5+1=6.

It happens that an expression contains parentheses within parentheses. There is no need to be afraid of this; you just need to consistently apply the stated rule for performing actions in expressions with brackets. Let's show the solution of the example.

Perform the operations in the expression 4+(3+1+4·(2+3)) .

This is an expression with brackets, which means that the execution of actions must begin with the expression in brackets, that is, with 3+1+4·(2+3) . This expression also contains parentheses, so you must perform the actions in them first. Let's do this: 2+3=5. Substituting the found value, we get 3+1+4·5. In this expression, we first perform multiplication, then addition, we have 3+1+4·5=3+1+20=24. The initial value, after substituting this value, takes the form 4+24, and all that remains is to complete the actions: 4+24=28.

In general, when an expression contains parentheses within parentheses, it is often convenient to perform actions starting with the inner parentheses and moving to the outer ones.

For example, let's say we need to perform the actions in the expression (4+(4+(4−6:2))−1)−1. First, we perform the actions in the inner brackets, since 4−6:2=4−3=1, then after this the original expression will take the form (4+(4+1)−1)−1. We again perform the action in the inner brackets, since 4+1=5, we arrive at the following expression (4+5−1)−1. Again we perform the actions in brackets: 4+5−1=8, and we arrive at the difference 8−1, which is equal to 7.

The order of operations in expressions with roots, powers, logarithms and other functions

If the expression includes powers, roots, logarithms, sine, cosine, tangent and cotangent, as well as other functions, then their values are calculated before performing other actions, and the rules from the previous paragraphs that specify the order of actions are also taken into account. In other words, the listed things, roughly speaking, can be considered enclosed in brackets, and we know that the actions in brackets are performed first.

Let's look at the solutions to the examples.

Perform the actions in the expression (3+1)·2+6 2:3−7.

This expression contains the power of 6 2, its value must be calculated before performing other actions. So, we perform the exponentiation: 6 2 =36. We substitute this value into the original expression, it will take the form (3+1)·2+36:3−7.

Then everything is clear: we perform the actions in brackets, after which we are left with an expression without brackets, in which, in order from left to right, we first perform multiplication and division, and then addition and subtraction. We have (3+1)·2+36:3−7=4·2+36:3−7= 8+12−7=13.

You can see other, including more complex examples of performing actions in expressions with roots, powers, etc., in the article Calculating the Values of Expressions.

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Online games, simulators, presentations, lessons, encyclopedias, articles

Examples with brackets, lesson with simulators.

We will look at three examples in this article:

1. Examples with parentheses (addition and subtraction actions)

2. Examples with parentheses (addition, subtraction, multiplication, division)

3. Examples with a lot of action

1 Examples with parentheses (addition and subtraction operations)

Let's look at three examples. In each of them, the order of actions is indicated by red numbers:

We see that the order of actions in each example will be different, although the numbers and signs are the same. This happens because there are parentheses in the second and third examples.

If there are no parentheses in the example, we perform all actions in order, from left to right.
If the example contains parentheses, then first we perform the actions in brackets, and only then all other actions, starting from left to right.

*This rule is for examples without multiplication and division. We will look at the rules for examples with parentheses involving the operations of multiplication and division in the second part of this article.

To avoid confusion in the example with parentheses, you can turn it into a regular example, without parentheses. To do this, write the result obtained in brackets above the brackets, then rewrite the entire example, writing this result instead of brackets, and then perform all the actions in order, from left to right:

In simple examples, you can perform all these operations in your mind. The main thing is to first perform the action in brackets and remember the result, and then count in order, from left to right.

And now - simulators!

1) Examples with brackets up to 20. Online simulator.

2) Examples with brackets up to 100. Online simulator.

3) Examples with brackets. Simulator No. 2

4) Insert the missing number - examples with brackets. Training apparatus

2 Examples with parentheses (addition, subtraction, multiplication, division)

Now let's look at examples in which, in addition to addition and subtraction, there is multiplication and division.

Let's look at examples without parentheses first:

If there are no parentheses in the example, first perform the operations of multiplication and division in order, from left to right. Then - the operations of addition and subtraction in order, from left to right.
If the example contains parentheses, then first we perform the operations in parentheses, then multiplication and division, and then addition and subtraction starting from left to right.

There is one trick to avoid getting confused when solving examples of the order of actions. If there are no parentheses, then we perform the operations of multiplication and division, then we rewrite the example, writing down the results obtained instead of these actions. Then we perform addition and subtraction in order:

If there are parentheses in the example, then you first need to get rid of the parentheses: rewrite the example, writing the result obtained in them instead of the parentheses. Then you need to mentally highlight the parts of the example, separated by the signs “+” and “-“, and count each part separately. Then perform addition and subtraction in order:

3 Examples with a lot of action

If there are many actions in the example, then it will be more convenient not to arrange the order of actions in the entire example, but to select blocks and solve each block separately. To do this, we find free signs “+” and “–” (free means not in brackets, shown in the figure with arrows).

These signs will divide our example into blocks:

When performing actions in each block, do not forget about the procedure given above in the article. Having solved each block, we perform the addition and subtraction operations in order.

Now let’s consolidate the solution to the examples on the order of actions on the simulators!

1. Examples with parentheses within numbers up to 100, addition, subtraction, multiplication and division. Online trainer.

2. Mathematics simulator for grades 2 - 3 “Arrange the order of actions (letter expressions).”

3. Order of actions (we arrange the order and solve examples)

Procedure in mathematics 4th grade

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 series of operations, you must first perform division or multiplication, then addition. 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 perform addition and subtraction, 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:

By doing of this assignment, first find the value of the expression enclosed in parentheses.
You should start with multiplication, then add.
After the expression in brackets is solved, we proceed to actions outside them.
According to the rules of procedure, next step there will be multiplication.
The final step will be subtraction.

As we see on clear 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. IN workbook corrections are not allowed. 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.

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Math lesson 2nd grade Order of actions in expressions with brackets.

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Target: 1.

3. Consolidate knowledge of the multiplication table and division by 2 – 6, the concept of divisor and

4. Learn to work in pairs in order to develop communication skills.

Equipment * : + — (), geometric material.

One, two - head up.

Three, four - arms wider.

Five, six - everyone sit down.

Seven, eight - let's discard laziness.

But first you have to find out its name. To do this you need to complete several tasks:

6 + 6 + 6 … 6 * 4 6 * 4 + 6… 6 * 5 – 6 14 dm 5 cm… 4 dm 5 cm

While we remembered the order of actions in expressions, miracles happened to the castle. We were just at the gate, and now we were in the corridor. Look, the door. And there is a castle on it. Shall we open it?

1. Subtract the quotient of 8 and 2 from the number 20.

2. Divide the difference between 20 and 8 by 2.

— How are the results different?

- Who can name the topic of our lesson?

(on massage mats)

Along the path, along the path

We gallop on our right leg,

We jump on our left leg.

Let's run along the path,

Our guess was completely correct7

Where are the actions performed first if there are parentheses in an expression?

Look at the “living examples” before us. Let's bring them to life.

* : + — ().

m – c * (a + d) + x

k: b + (a – c) * t

6. Work in pairs.

To solve them you will need geometric material.

Students complete tasks in pairs. After completion, check the work of the pairs at the board.

What new have you learned?

8. Homework.

Topic: Order of actions in expressions with brackets.

Target: 1. Derive a rule for the order of actions in expressions with brackets containing all

4 arithmetic operations,

2. Form the ability to practical application rules,

4. Learn to work in pairs in order to develop communication skills.

Equipment: textbook, notebooks, cards with action signs * : + — (), geometric material.

1 .Physical exercise.

Nine, ten - sit down quietly.

2. Updating basic knowledge.

Today we are setting off on another journey through the Land of Knowledge, the city of mathematics. We have to visit one palace. Somehow I forgot its name. But let’s not be upset, you yourself can tell me its name. While I was worried, we approached the gates of the palace. Shall we come in?

1. Compare expressions:

2. Unscramble the word.

3. Statement of the problem. Discovery of something new.

So what is the name of the palace?

And when in mathematics do we talk about order?

What do you already know about the order of actions in expressions?

— Interesting, we are asked to write down and solve expressions (the teacher reads the expressions, the students write them down and solve them).

20 – 8: 2

(20 – 8) : 2

Well done. What's interesting about these expressions?

Look at the expressions and their results.

— What is common in writing expressions?

- Why do you think it turned out? different results, because the numbers were the same?

Who would dare to formulate a rule for performing actions in expressions with brackets?

We can check the correctness of this answer in another room. Let's go there.

4. Physical exercise.

And along the same path

We will reach the mountain.

Stop. Let's rest a little

And we'll go on foot again.

5. Primary consolidation of what has been learned.

Here we are.

We need to solve two more expressions to check the correctness of our assumption.

6 * (33 – 25) 54: (6 + 3) 25 – 5 * (9 – 5) : 2

To check the correctness of the assumption, let's open the textbooks on page 33 and read the rule.

How should you perform the actions after the solution in brackets?

Letter expressions are written on the board and there are cards with action signs. * : + — (). Children go to the board one at a time, take a card with the action that needs to be done first, then the second student comes out and takes a card with the second action, etc.

a + (a – b)

a * (b + c) : d – t

m – c * ( a + d ) + x

k : b + ( a – c ) * t

(a–b) : t+d

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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 an 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.

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 brackets.
You should start with multiplication, then add.
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.

Composing an Expression with Parentheses

1. Make up expressions with brackets from the following sentences and solve them.

From the number 16, subtract the sum of the numbers 8 and 6.
From the number 34, subtract the sum of the numbers 5 and 8.
Subtract the sum of numbers 13 and 5 from the number 39.
The difference between the numbers 16 and 3 is added to the number 36
Add the difference between 48 and 28 to 16.

2. Solve the problems by first composing the correct expressions, and then solving them sequentially:

2.1. Dad brought a bag of nuts from the forest. Kolya took 25 nuts from the bag and ate them. Then Masha took 18 nuts from the bag. Mom also took 15 nuts from the bag, but put 7 of them back. How many nuts are left in the bag in the end if there were 78 of them at the beginning?

2.2. The foreman was repairing parts. At the beginning of the workday there were 38 of them. In the first half of the day he was able to repair 23 of them. In the afternoon they brought him the same amount as they had at the very beginning of the day. In the second half, he repaired another 35 parts. How many parts does he have left to repair?

3. Solve the examples correctly following the sequence of actions:

45: 5 + 12 * 2 -21:3
56 - 72: 9 + 48: 6 * 3
7 + 5 * 4 - 12: 4
18: 3 - 5 + 6 * 8

Solving expressions with parentheses

1. Solve the examples by opening the brackets correctly:

1 + (4 + 8) =	8 - (2 + 4) =	3 + (6 - 5) =	59 + 25 =
82 + 14 =	29 + 52 =	18 + 47 =	39 + 53 =
37 + 53 =	25 + 63 =	87 + 17 =	19 + 52 =

2. Solve the examples correctly following the sequence of actions:

2.1. 36: 3 + 12 * (2 - 1) : 3
2.2. 39 - (81: 9 + 48: 6) * 2
2.3. (7 + 5) * 2 - 48: 4
2.4. 18: 3 + (5 * 6) : 2 - 4

3. Solve the problems by first composing the correct expressions, and then solving them sequentially:

3.1. There were 25 packages in stock washing powder. 12 packages were taken to one store. Then the same amount was taken to the second store. After that they brought it to the warehouse 3 times more packages than it was before. How many packages of powder are in stock?

3.2. There were 75 tourists staying at the hotel. On the first day, 3 groups of 12 people each left the hotel, and 2 groups of 15 people each arrived. On the second day, another 34 people left. How many tourists remained in the hotel at the end of 2 days?

3.3. They brought 2 bags of clothes to the dry cleaner, 5 items in each bag. Then they took 8 things. In the afternoon they brought 18 more items for washing. And they only took 5 washed items. How many items are in the dry cleaner at the end of the day if there were 14 items at the beginning of the day?

FI _________________________________

21: 3 * 6 - (18 + 14) : 8 =	63: (81: 9) + (8 * 7 - 2) : 6 =	64:2: 4+ 97-91=
37 *2 + 180: 9 – 36: 12 =	52 * 10 – 60: 15 * 1 =	72: 4 +58:2=
5 *0: 25 + (72: 1 – 0) : 9 =	21: (3 * 7) – (7* 0 + 1)*1 =	6:6+0:8-8:8=
91: 7 + 80: 5 – 5: 5 =	64:4 - 3*5 +80:2=	(19*5 – 5) : 30 =
19 + 17 * 3 – 46 =	(39+29) : 4 + 8*0=	(60-5) : 5 +80: 5=
54 – 26 + 38: 2 =	63: (73) 3=	(160-70) : 18 *1=
200 – 80: 5 + 3 * 4 =	(29+25): (72:8)=	72:25 + 3* 17=
80: 16 + 660: 6 =	3 * 290 – 800=	950:50*1-0=
(48: 3) : 16 * 0 =	90-6*6+29=	5* (48-43) +15:5*7=
54: 9 8 - 14: 7 4 =	63: 74+70:7 5=	24: 67 - 70=
21: 7 * 8 + 32: 8 * 4 =	27: 3* 5 + 26-18 *4=	54: 6*7 - 0:1=
45: 9 * 6 + 7 * 5 – 26 =	28: 7 9 + 6 (54 – 47)=	6*(9: 3) - 40:5 =
21 * 1 - 56: 7 – 8 =	9 * (64: 8) - 18:18	3 *(14: 2) - 63:9=
4 * 8 + 42: 6 *5 =	04+0:5 +8 (48: 8)=	56:7 +76 - 51=
31 * 3 - 17 – 80: 16 * 1 =	57:19 32 - 11 7=	72-96:8 +60:15 *13=
36 + 42: 3 + 23 + 27 *0 =	56:14 *19 - 72:18=	(86-78:13)* 4=
650 – 50 * 4 + 900: 100 =	630: 9 + 120 * 5 + 40=	980 – (160 + 20) : 30=
940 - (1680 – 1600) * 9 =	29* 2+26 – 37:2=	72:3 +280: (14*5)=
300: (5 60) (78: 13) =	63+ 100: 4 – 8*0=	84:7+70:14 – 6:6=
45: 15 – 180: 90 + 84: 7 =	32+51 + 48:6 * 5=	54:6 ?2 – 70:14=
38: 2 – 48: 3 + 0 * 9 =	30:6 * 8 – 6+3*2=	(95:19) *(68:2)=
(300 - 8 * 7) * 10 =	1:1 - 00 + 10 - 1*1=	(80: 4 – 60:30) *5 =
2 * (120: 6 – 80: 20) =	56:4+96:3- 0*7=	20+ 20: 4 - 1*5=
(18 + 14) : 8 – (7 0 + 1) 1 =	(87-2):6 +63: (73)=	(50-5) : 5+21: (3*7)=
19 + 17 * 3 – 60: 15 * 1 =	80: 5 +3*5 +80:2=	54: 9 8-64:4 +160=
72 * 10 - 64: 2: 4 =	84 – 36 + 38:2	91:13+80:5 – 5:5
300 – 80: 5 + 6 * 4 =	950:190 1+14: 74=	(39+29) : 17 + 8*0=
(120 - 30) : 18 * 1- 72: 25 =	210:30*60-0:1=	90-67+3 17=
240: 60 7 – 7 0 =	60:60+0:80-80:80=	720: 40 +580:20=
9 7 – 9 1 + 5 * 0: 25 =	21: 7 * 6 +32: 4 *5=	80:16 +66:6 -63:(81:9)=
(19 * 5 – 5) : 30 + 70: 7 =	15:57 + 63: 7 5=	54: 6 * 7 - (72:1-0):9=
3 290 – 600 – 5 (48 – 43) =	(300-897)10 - 3?2=	(80: 4) +30*2+ 180: 9=
30: 6 * 8 – 6 + 48: 3 + 0 *9 =	(95:19) (68:34) - 60:305=	27: 3*5 - 48:3=
3* 290 – 800 + 950: 50 =	80:16 +660:6*1-0=	90-66+ 15:57=
5(48 - 43) + (48: 3) :160=	280: (145) +630: 90=	300: (506) (78: 6)=

If in the examples you encounter question mark(?), it should be replaced with the sign * - multiplication.

1. SOLVE EXPRESSIONS:

35: 5 + 36: 4 - 3
26 + 6 x 8 – 45: 5 24: 6 + 18 – 2 x 6
9 x 6 – 3 x 6 + 19 – 27:3

2. SOLVE EXPRESSIONS:

48: 8 + 32 – 54: 6 + 7 x 4
17 + 24: 3 x 4 – 27: 3 x 2 6 x 4: 3 + 54: 6: 3 x 6 + 2 x 9
100 – 6 x 2: 3 x 9 – 39 + 7 x 4

3. SOLVE EXPRESSIONS:

100 – 27: 3 x 6 + 7 x 4
2 x 4 + 24: 3 + 18: 6 x 9 9 x 3 – 19 + 6 x 7 – 3 x 5
7 x 4 + 35: 7 x 5 – 16: 2: 4 x 3

4. SOLVE EXPRESSIONS:

32: 8 x 6: 3 + 6 x 8 – 17
5 x 8 – 4 x 7 + 13 - 11 24: 6 + 18: 2 + 20 – 12 + 6 x 7
21: 3 – 35: 7 + 9 x 3 + 9 x 5

5. SOLVE EXPRESSIONS:

42: 7 x 3 + 2 + 24: 3 – 7 + 9 x 3
6 x 6 + 30: 5: 2 x 7 - 19 90 - 7 x 5 – 24: 3 x 5
6 x 5 – 12: 2 x 3 + 49

6. SOLVE EXPRESSIONS:

32: 8 x 7 + 54: 6: 3 x 5
50 – 45: 5 x 3 + 16: 2 x 5 8 x 6 + 23 – 24: 4 x 3 + 17
48: 6 x 4 + 6 x 9 – 26 + 13

7. SOLVE EXPRESSIONS:

42: 6 + (19 + 6) : 5 – 6 x 2
60 – (13 + 22) : 5 – 6 x 4 + 25 (27 – 19) x 4 + 18: 3 + (8 + 27) :5 -17
(82 – 74) : 2 x 7 + 7 x 4 - (63 – 27): 4
8. SOLVE EXPRESSIONS:

90 – (40 – 24: 3) : 4 x 6 + 3 x 5
3 x 4 + 9 x 6 – (27 + 9) : 4 x 5
(50 – 23) : 3 + 8 x 5 – 6 x 5 + (26 + 16) : 6
(5 x 6 – 3 x 4 + 48: 6) + (82 – 78) x 7 – 13
54: 9 + (8 + 19) : 3 – 32: 4 – 21: 7 + (42 – 14) : 4 – (44 14) : 5

9. SOLVE EXPRESSIONS:

9 x 6 – 6 x 4: (33 – 25) x 7
3 x (12 – 8) : 2 + 6 x 9 - 33 (5 x 9 - 25) : 4 x 8 – 4 x 7 + 13
9 x (2 x 3) – 48: 8 x 3 + 7 x 6 - 34

10. SOLVE EXPRESSIONS:

(8 x 6 – 36:6) : 6 x 3 + 5 x 9
7 x 6 + 9 x 4 – (2 x 7 + 54: 6 x 5) (76 – (27 + 9) + 8) : 6 x 4
(7 x 4 + 33) – 3 x 6:2

11. SOLVE EXPRESSIONS:

(37 + 7 x 4 – 17) : 6 + 7 x 5 + 33 + 9 x 3 – (85 – 67) : 2 x 5
5 x 7 + (18 + 14) : 4 – (26 – 8) : 3 x 2 – 28: 4 + 27: 3 – (17 + 31) : 6

12. SOLVE EXPRESSIONS:

(58 – 31) : 3 – 2 + (58 – 16) : 6 + 8 x 5 – (60 – 42) : 3 + 9 x 2
(9 x 7 + 56: 7) – (2 x 6 – 4) x 3 + 54: 9

13. SOLVE EXPRESSIONS:

(8 x 5 + 28: 7) + 12: 2 – 6 x 5 + (13 – 5) x 4 + 5 x 4
(7 x 8 – 14:7) + (7 x 4 + 12:6) – 10:5 + 63:9

Test “Order of arithmetic operations” (1 option)
1(1b)
2(1b)
3(1b)
4(3b)
5(2b)
6(2b)
7(1b)
8(1b)
9(3b)
10(3b)
11(3b)
12(3b)

110 – (60 +40) :10 x 8

a) 800 b) 8 c) 30

a) 3 4 6 5 2 1 4 5 6 3 2 1

3 4 6 5 1 2

5. In which of the expressions is the last action multiplication?
a) 1001:13 x (318 +466) :22

c) 10000 – (5 x 9+56 x 7) x2
6. In which of the expressions is the first action subtraction?
a) 2025:5 – (524 – 24:6) x45
b) 5870 + (90-50 +30) x8 -90
c) 5400:60 x (3600:90 -90)x5

Choose the correct answer:
9. 90 – (50- 40:5) x 2+ 30
a) 56 b) 92 c) 36
10. 100- (2x5+6 - 4x4) x2
a) 100 b) 200 c) 60
11. (10000+10000:100 +400) : 100 +100
a) 106 b) 205 c) 0
12. 150: (80 – 60:2) x 3
a) 9 b) 45 c) 1

Test "Order of Arithmetic Operations"
1(1b)
2(1b)
3(1b)
4(3b)
5(2b)
6(2b)
7(1b)
8(1b)
9(3b)
10(3b)
11(3b)
12(3b)
1. Which action in the expression will you do first?
560 – (80+20) :10 x7
a) addition b) division c) subtraction
2. What action in the same expression will you do second?
a) subtraction b) division c) multiplication
3. Choose the correct answer to this expression:
a) 800 b) 490 c) 30
4. Choose the correct arrangement of actions:
a) 3 4 6 5 2 1 4 5 6 3 2 1
320: 8 x 7 + 9 x (240 – 60:15) c) 320: 8 x 7 + 9x (240 – 60:15)

3 4 6 5 2 1
b) 320: 8 x 7 + 9 x (240 – 60:15)
5. In which of the expressions is the last action division?
a) 1001:13 x (318 +466) :22
b) 391 x37:17 x (2248:8 – 162)
c) 10000 – (5 x 9+56 x 7) x2
6. In which of the expressions is the first action addition?
a) 2025:5 – (524 + 24 x6) x45
b) 5870 + (90-50 +30) x8 -90
c) 5400:60 x (3600:90 -90)x5
7. Choose the correct statement: “In an expression without parentheses, the actions are performed:”
a) in order b) x and: , then + and - c) + and -, then x and:
8. Choose the correct statement: “In an expression with brackets, the actions are performed:”
a) first in brackets b)x and:, then + and - c) in writing order
Choose the correct answer:
9. 120 – (50- 10:2) x 2+ 30
a) 56 b) 0 c) 60
10. 600- (2x5+8 - 4x4) x2
a) 596 b) 1192 c) 60
11. (20+20000:2000 +30) : 20 +200
a) 106 b) 203 c) 0
12. 160: (80 – 80:2) x 3
a) 120 b) 0 c) 1

When we work with various expressions that include numbers, letters and variables, we have to perform a large number of arithmetic operations. When we do a conversion or calculate a value, it is very important to follow the correct order of these actions. In other words, arithmetic operations have their own special order of execution.

Yandex.RTB R-A-339285-1

In this article we will tell you which actions should be done first and which ones after. First, let's look at a few simple expressions that contain only variables or numeric values, as well as division, multiplication, subtraction and addition signs. Then let's take examples with parentheses and consider in what order they should be calculated. In the third part we will give the necessary order of transformations and calculations in those examples that include signs of roots, powers and other functions.

Definition 1

In the case of expressions without parentheses, the order of actions is determined unambiguously:

All actions are performed from left to right.
We perform division and multiplication first, and subtraction and addition second.

The meaning of these rules is easy to understand. The traditional left-to-right writing order defines the basic sequence of calculations, and the need to multiply or divide first is explained by the very essence of these operations.

Let's take a few tasks for clarity. We used only the simplest numerical expressions so that all calculations could be done mentally. This way you can quickly remember the desired order and quickly check the results.

Example 1

Condition: calculate how much it will be 7 − 3 + 6 .

Solution

There are no parentheses in our expression, there is also no multiplication and division, so we perform all the actions in the specified order. First we subtract three from seven, then add six to the remainder and end up with ten. Here is a transcript of the entire solution:

7 − 3 + 6 = 4 + 6 = 10

Answer: 7 − 3 + 6 = 10 .

Example 2

Condition: in what order should the calculations be performed in the expression? 6:2 8:3?

Solution

To answer this question, let’s reread the rule for expressions without parentheses that we formulated earlier. We only have multiplication and division here, which means we keep the written order of calculations and count sequentially from left to right.

Answer: First we divide six by two, multiply the result by eight and divide the resulting number by three.

Example 3

Condition: calculate how much it will be 17 − 5 · 6: 3 − 2 + 4: 2.

Solution

First, let's determine the correct order of operations, since we have all the basic types of arithmetic operations here - addition, subtraction, multiplication, division. The first thing we need to do is divide and multiply. These actions do not have priority over each other, so we perform them in the written order from right to left. That is, 5 must be multiplied by 6 to get 30, then 30 divided by 3 to get 10. After that, divide 4 by 2, this is 2. Let's substitute the found values into the original expression:

17 − 5 6: 3 − 2 + 4: 2 = 17 − 10 − 2 + 2

There is no longer division or multiplication here, so we do the remaining calculations in order and get the answer:

17 − 10 − 2 + 2 = 7 − 2 + 2 = 5 + 2 = 7

Answer:17 − 5 6: 3 − 2 + 4: 2 = 7.

Until the order of performing actions is firmly memorized, you can put numbers above the signs of arithmetic operations indicating the order of calculation. For example, for the problem above we could write it like this:

If we have letter expressions, then we do the same with them: first we multiply and divide, then we add and subtract.

What are the first and second stage actions?

Sometimes in reference books all arithmetic operations are divided into actions of the first and second stages. Let us formulate the necessary definition.

The operations of the first stage include subtraction and addition, the second - multiplication and division.

Knowing these names, we can write the previously given rule regarding the order of actions as follows:

Definition 2

In an expression that does not contain parentheses, you must first perform the actions of the second stage in the direction from left to right, then the actions of the first stage (in the same direction).

Order of calculations in expressions with parentheses

The parentheses themselves are a sign that tells us the desired order of actions. In this case the right rule can be written like this:

Definition 3

If there are parentheses in the expression, then the first step is to perform the operation in them, after which we multiply and divide, and then add and subtract from left to right.

As for the parenthetical expression itself, it can be considered as an integral part of the main expression. When calculating the value of the expression in brackets, we maintain the same procedure known to us. Let's illustrate our idea with an example.

Example 4

Condition: calculate how much it will be 5 + (7 − 2 3) (6 − 4) : 2.

Solution

There are parentheses in this expression, so let's start with them. First of all, let's calculate how much 7 − 2 · 3 will be. Here we need to multiply 2 by 3 and subtract the result from 7:

7 − 2 3 = 7 − 6 = 1

We calculate the result in the second brackets. There we have only one action: 6 − 4 = 2 .

Now we need to substitute the resulting values into the original expression:

5 + (7 − 2 3) (6 − 4) : 2 = 5 + 1 2: 2

Let's start with multiplication and division, then perform subtraction and get:

5 + 1 2: 2 = 5 + 2: 2 = 5 + 1 = 6

This concludes the calculations.

Answer: 5 + (7 − 2 3) (6 − 4) : 2 = 6.

Don't be alarmed if our condition contains an expression in which some parentheses enclose others. We only need to apply the rule above consistently to all expressions in parentheses. Let's take this problem.

Example 5

Condition: calculate how much it will be 4 + (3 + 1 + 4 (2 + 3)).

Solution

We have parentheses within parentheses. We start with 3 + 1 + 4 · (2 + 3), namely 2 + 3. It will be 5. The value will need to be substituted into the expression and calculated that 3 + 1 + 4 · 5. We remember that we first need to multiply and then add: 3 + 1 + 4 5 = 3 + 1 + 20 = 24. Substituting the found values into the original expression, we calculate the answer: 4 + 24 = 28 .

Answer: 4 + (3 + 1 + 4 · (2 + 3)) = 28.

In other words, when calculating the value of an expression that includes parentheses within parentheses, we start with the inner parentheses and work our way to the outer ones.

Let's say we need to find how much (4 + (4 + (4 − 6: 2)) − 1) − 1 will be. We start with the expression in the inner brackets. Since 4 − 6: 2 = 4 − 3 = 1, the original expression can be written as (4 + (4 + 1) − 1) − 1. Looking again at the inner parentheses: 4 + 1 = 5. We have come to the expression (4 + 5 − 1) − 1 . We count 4 + 5 − 1 = 8 and as a result we get the difference 8 - 1, the result of which will be 7.

The order of calculation in expressions with powers, roots, logarithms and other functions

If our condition contains an expression with a degree, root, logarithm or trigonometric function(sine, cosine, tangent and cotangent) or other functions, then first of all we calculate the value of the function. After this, we act according to the rules specified in the previous paragraphs. In other words, functions are equal in importance to the expression enclosed in brackets.

Let's look at an example of such a calculation.

Example 6

Condition: find how much is (3 + 1) · 2 + 6 2: 3 − 7.

Solution

We have an expression with a degree, the value of which must be found first. We count: 6 2 = 36. Now let’s substitute the result into the expression, after which it will take the form (3 + 1) · 2 + 36: 3 − 7.

(3 + 1) 2 + 36: 3 − 7 = 4 2 + 36: 3 − 7 = 8 + 12 − 7 = 13

Answer: (3 + 1) 2 + 6 2: 3 − 7 = 13.

In a separate article devoted to calculating the values of expressions, we provide other, more complex examples of calculations in the case of expressions with roots, degrees, etc. We recommend that you familiarize yourself with it.

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The video tutorial “Procedure for performing actions” explains in detail important topic mathematics - the sequence of performing arithmetic operations when solving an expression. During the video lesson, it is discussed what priority various mathematical operations have, how they are used in calculating expressions, examples are given for mastering the material, and the knowledge gained is generalized in solving tasks where all the considered operations are present. With the help of a video lesson, the teacher has the opportunity to quickly achieve the goals of the lesson and increase its effectiveness. The video can be used as visual material to accompany the teacher’s explanation, as well as as an independent part of the lesson.

Visual material uses techniques that help to better understand the topic, as well as remember important rules. With the help of color and different writing, the features and properties of operations are highlighted, and the peculiarities of solving examples are noted. Animation effects help deliver consistency educational material and also draw students' attention to important points. The video is voiced, so it is supplemented with comments from the teacher, helping the student understand and remember the topic.

The video lesson begins by introducing the topic. Then it is noted that multiplication and subtraction are operations of the first stage, operations of multiplication and division are called operations of the second stage. This definition will need to be operated further, displayed on the screen and highlighted in large color font. Then the rules that make up the order of operations are presented. The first order rule is derived, which indicates that if there are no parentheses in the expression, and there are actions of the same level, these actions must be performed in order. The second order rule states that if there are actions of both stages and there are no parentheses, the operations of the second stage are performed first, then the operations of the first stage are performed. The third rule sets the order of operations for expressions that include parentheses. It is noted that in this case the operations in brackets are performed first. The wording of the rules is highlighted in colored font and is recommended for memorization.

Next, it is proposed to understand the order of operations by considering examples. The solution to an expression containing only addition and subtraction operations is described. The main features that affect the order of calculations are noted - there are no parentheses, there are first-stage operations. Below is a description of how calculations are performed, first subtraction, then addition twice, and then subtraction.

In the second example 780:39·212:156·13 you need to evaluate the expression, performing actions according to the order. It is noted that this expression contains exclusively second-stage operations, without parentheses. IN in this example all actions are performed strictly from left to right. Below we describe the actions one by one, gradually approaching the answer. The result of the calculation is the number 520.

The third example considers a solution to an example in which there are operations of both stages. It is noted that in this expression there are no parentheses, but there are actions of both stages. According to the order of operations, the second stage operations are performed, followed by the first stage operations. Below is a step-by-step description of the solution, in which three operations are performed first - multiplication, division, and another division. Then, first-stage operations are performed with the found values of the product and quotients. During the solution, the actions of each step are combined in curly braces for clarity.

The following example contains parentheses. Therefore, it is demonstrated that the first calculations are performed on the expressions in parentheses. After them, the second stage operations are performed, followed by the first.

The following is a note about in what cases you can not write parentheses when solving expressions. It is noted that this is only possible in the case where eliminating the parentheses does not change the order of operations. An example is the expression with brackets (53-12)+14, which contains only first-stage operations. Having rewritten 53-12+14 with the elimination of parentheses, you can note that the order in which the value is searched will not change - first the subtraction 53-12=41 is performed, and then the addition 41+14=55. It is noted below that you can change the order of operations when finding a solution to an expression using the properties of the operations.

At the end of the video lesson, the material studied is summarized in the conclusion that each expression requiring a solution specifies a specific program for calculation, consisting of commands. An example of such a program is presented in the description of the solution complex example, which is the quotient of (814+36·27) and (101-2052:38). The given program contains the following points: 1) find the product of 36 with 27, 2) add the found sum to 814, 3) divide the number 2052 by 38, 4) subtract the result of dividing 3 points from the number 101, 5) divide the result of step 2 by the result of point 4.

At the end of the video lesson there is a list of questions that students are asked to answer. These include the ability to distinguish between actions of the first and second stages, questions about the order of actions in expressions with actions of the same stage and different stages, about the order of actions in the presence of parentheses in the expression.

The video tutorial “Order of Actions” is recommended to be used on a traditional school lesson to increase the effectiveness of the lesson. Also visual material will be useful for distance learning. If a student needs an additional lesson to master a topic or is studying it independently, the video can be recommended for independent study.