We will look at three examples in this article:

1. Examples with brackets (addition and subtraction operations)

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

3. Examples with a lot of actions

1 Examples with brackets (addition and subtraction operations)

Let's look at three examples. In each of them, the procedure 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 is because the second and third examples have parentheses.

*This rule is for examples without multiplication and division. Rules for examples with brackets, including the operations of multiplication and division, we will consider in the second part of this article.

In order not to get confused in the example with brackets, you can turn it into a regular example, without brackets. To do this, we write the result obtained in brackets above the brackets, then we rewrite the entire example, writing this result instead of brackets, and then we perform all the actions in order, from left to right:

In simple examples, all these operations can be performed in the 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.

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

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

3) Examples with brackets. Trainer #2

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

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

Now consider examples in which, in addition to addition and subtraction, there is multiplication and division.

Let's look at examples without parentheses first:

There is one trick, how not to get confused when solving examples for the order of actions. If there are no brackets, 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 the example contains brackets, then first you need to get rid of the brackets: rewrite the example, writing the result obtained in them instead of brackets. 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 the free signs "+" and "-" (free means not in brackets, shown by arrows in the figure).

These signs will divide our example into blocks:

Performing the actions in each block, do not forget about the procedure given above in the article. After solving each block, we perform addition and subtraction operations in order.

And now we fix the solution of the examples on the order of actions on the simulators!

If games or simulators do not open for you, read.

To correctly evaluate expressions in which you need to perform more than one operation, you need to know the order in which arithmetic operations are performed. Arithmetic operations in the expression without brackets agreed to be performed in the following order:

1. If there is exponentiation in the expression, then this action is first performed in sequential order, that is, from left to right.
2. Then (if present in the expression), the operations of multiplication and division are performed in the order in which they appear.
3. The last (if present in the expression) operations of addition and subtraction are performed in the order in which they appear.

As an example, consider the following expression:

First you need to perform exponentiation (square the number 4 and cube the number 2):

3 16 - 8: 2 + 20

Then multiplication and division are performed (3 times 16 and 8 divided by 2):

And at the very end, subtraction and addition are performed (subtract 4 from 48 and add 20 to the result):

48 - 4 + 20 = 44 + 20 = 64

Steps 1 and 2

Arithmetic operations are divided into operations of the first and second stages. Addition and subtraction are called first step actions, multiplication and division - second step actions.

If the expression contains actions of only one stage and there are no brackets in it, then the actions are performed in the order they appear from left to right.

Example 1

15 + 17 - 20 + 8 - 12

Solution. This expression contains the actions of only one stage - the first (addition and subtraction). It is necessary to determine the order of actions and carry them out.

Answer: 42.

If the expression contains the actions of both stages, then the actions of the second stage are executed first, in their order (from left to right), and then the actions of the first stage.

Example. Calculate the value of an expression:

24:3 + 5 2 - 17

Solution. This expression contains four actions: two of the first stage and two of the second. Let's define the order of their execution: according to the rule, the first action will be division, the second - multiplication, the third - addition, and the fourth - subtraction.

Now let's start the calculation.

When we work with various expressions, including numbers, letters and variables, we have to perform a large number of arithmetic operations. When we do a transformation 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 execution order.

In this article, we will tell you what actions should be done first and which 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 we will take examples with brackets and consider in what order they should be evaluated. In the third part, we will give the correct order of transformations and calculations in those examples that include the signs of roots, powers, and other functions.

Definition 1

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

1. All actions are performed from left to right.
2. First of all, we perform division and multiplication, and secondly, subtraction and addition.

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 have used only the simplest numerical expressions so that all calculations can be done mentally. So you can quickly remember the desired order and quickly check the results.

Example 1

Condition: calculate how much 7 − 3 + 6 .

Solution

There are no brackets in our expression, multiplication and division are also absent, so we perform all the actions in the specified order. First, subtract three from seven, then add six to the remainder, and as a result we get ten. Here is a record 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, we reread the rule for expressions without parentheses, which 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 will be 17 − 5 6: 3 − 2 + 4: 2.

Solution

First, let's determine the correct order of operations, since we have here all the basic types of arithmetic operations - 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 and get 30, then 30 divided by 3 and get 10. After that we divide 4 by 2 , that's 2 . Substitute the found values ​​into the original expression:

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

There is no 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 learned, you can put numbers over 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 literal expressions, then we do the same with them: first we multiply and divide, then we add and subtract.

What are steps one and two

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

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

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

Definition 2

In an expression that doesn't have parentheses, you must first perform the actions of the second step in the direction from left to right, then the actions of the first step (in the same direction).

Order of evaluation in expressions with brackets

Parentheses themselves are a sign that tells us the desired order in which to perform actions. In this case, the desired rule can be written as follows:

Definition 3

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

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

Example 4

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

Solution

This expression has parentheses, 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 consider 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 subtract and get:

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

This completes the calculations.

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

Do not be alarmed if the condition contains an expression in which some brackets enclose others. We only need to apply the rule above consistently to all parenthesized expressions. Let's take this task.

Example 5

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

Solution

We have brackets within brackets. 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 calculate that 3 + 1 + 4 5 . We remember that we must first 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 evaluating the value of an expression involving 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 will be (4 + (4 + (4 - 6: 2)) - 1) - 1. 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 . Again we turn to the inner brackets: 4 + 1 = 5 . We have come to the expression (4 + 5 − 1) − 1 . We believe 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 we have an expression in the condition 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 that, 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 will be (3 + 1) 2 + 6 2: 3 - 7 .

Solution

We have an expression with a degree, the value of which must be found first. We consider: 6 2 \u003d 36. Now we 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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In this lesson, the procedure for performing arithmetic operations in expressions without brackets and with brackets is considered in detail. Students are given the opportunity in the course of 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 differs in expressions without brackets and with brackets, to practice applying the learned rule, to find and correct errors made in determining the order of actions.

In life, we constantly perform some kind of action: we walk, study, read, write, count, smile, quarrel and make up. We perform these steps in a different order. Sometimes they can be swapped, sometimes they can't. For example, going to school in the morning, you can first do exercises, then make the bed, or vice versa. But you can’t go to school first and then put on clothes.

And 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 execute actions in one expression from left to right, and in another from right to left. Numbers can indicate the order in which actions are performed (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 result 7 from 8.

We see that the values ​​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 brackets.

If the expression without brackets 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 has only addition and subtraction operations. These actions are called first step actions.

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

Rice. 2. Procedure

Consider the second expression

In this expression, there are only operations of multiplication and division - These are the second step actions.

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

Consider an expression.

We reason 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 lay out the procedure.

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 the expression contains parentheses?

If the expression contains parentheses, then the value of the expressions in the parentheses is calculated first.

Consider an expression.

30 + 6 * (13 - 9)

We see that in this expression there is an action in brackets, which means that we will perform this action first, then, in order, multiplication and addition. Let's lay out the procedure.

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 in order to correctly establish the order of arithmetic operations in a numerical expression?

Before proceeding with the calculations, it is necessary to consider the expression (find out if it contains brackets, what actions it has) and only after that 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.

Consider the expressions, establish the order of operations and perform the calculations.

43 - (20 - 7) +15

32 + 9 * (19 - 16)

Let's follow the rules. The expression 43 - (20 - 7) +15 has operations in parentheses, as well as operations of addition and subtraction. Let's set the course of action. The first step is to perform the action in brackets, 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) has operations in parentheses, as well as operations of multiplication and addition. According to the rule, we first perform the action in brackets, then multiplication (the number 9 is multiplied 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 brackets, but there are operations of multiplication, division and subtraction. We act according to the rule. First, we perform multiplication and division from left to right, and then from the result obtained by multiplication, we subtract the result obtained by division. 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 if the order of actions in the following expressions is defined correctly.

37 + 9 - 6: 2 * 3 =

18: (11 - 5) + 47=

7 * 3 - (16 + 4)=

We reason like this.

37 + 9 - 6: 2 * 3 =

There are no brackets 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 order of actions is defined correctly.

Find the value of this expression.

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

We continue to argue.

The second expression contains brackets, which means that we first perform the action in brackets, then from left to right multiplication or division, addition or subtraction. We check: the first action is in brackets, the second is division, the third is addition. Conclusion: the order of actions is defined incorrectly. Correct the errors, find the value of the expression.

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

This expression also contains brackets, which means that we first perform the action in brackets, then from left to right multiplication or division, addition or subtraction. We check: the first action is in brackets, the second is multiplication, the third is subtraction. Conclusion: the order of actions is defined incorrectly. Correct the errors, find the value 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 studied rule (Fig. 5).

Rice. 5. Procedure

We do not see numerical values, so we will not be able to find the meaning of expressions, but we will practice applying the learned rule.

We act according to the algorithm.

The first expression has parentheses, so 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 brackets, which means that we perform the first action in brackets. After that, from left to right, multiplication and division, after that - subtraction.

Let's check ourselves (Fig. 6).

Rice. 6. Procedure

Today in the lesson we got acquainted with the rule of the order of execution of actions in expressions without brackets and with brackets.

Bibliography

1. M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Enlightenment", 2012.
2. M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Enlightenment", 2012.
3. M.I. Moreau. Mathematics lessons: Guidelines for teachers. Grade 3 - M.: Education, 2012.
4. Regulatory document. Monitoring and evaluation of learning outcomes. - M.: "Enlightenment", 2011.
5. "School of Russia": Programs for elementary school. - M.: "Enlightenment", 2011.
6. S.I. Volkov. Mathematics: Testing work. Grade 3 - M.: Education, 2012.
7. V.N. Rudnitskaya. Tests. - M.: "Exam", 2012.

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

Homework

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

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

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

3. Compose 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.

Today we will talk about execution order mathematical action. What action to take first? Addition and subtraction, or multiplication and division. Strangely, our children have trouble solving seemingly elementary expressions.

So, remember that the expressions in brackets are evaluated first.

38 – (10 + 6) = 22 ;

Order of actions:

1) in brackets: 10 + 6 = 16;

2) subtraction: 38 - 16 \u003d 22.

If the expression without brackets includes only addition and subtraction, or only multiplication and division, then the operations are performed in order from left to right.

10 ÷ 2 × 4 = 20 ;

Order of actions:

1) from left to right, division first: 10 ÷ 2 = 5;

2) multiplication: 5 × 4 = 20;

10 + 4 - 3 \u003d 11, i.e.:

1) 10 + 4 = 14 ;

2) 14 – 3 = 11 .

If in an expression without brackets there is not only addition and subtraction, but also multiplication or division, then the actions are performed in order from left to right, but multiplication and division have the advantage, they are performed first, followed by addition and subtraction.

18 ÷ 2 - 2 × 3 + 12 ÷ 3 = 7

The order of actions:

1) 18 ÷ 2 = 9;

2) 2 × 3 = 6;

3) 12 ÷ 3 = 4;

4) 9 - 6 = 3; those. from left to right - the result of the first action minus the result of the second;

5) 3 + 4 = 7; those. the result of the fourth action plus the result of the third;

If the expression contains parentheses, then the expressions in the parentheses are executed first, then the multiplication and division, and only then the addition and subtraction.

30 + 6 × (13 - 9) \u003d 54, i.e.:

1) expression in brackets: 13 - 9 = 4;

2) multiplication: 6 × 4 = 24;

3) addition: 30 + 24 = 54;

So, let's sum up. Before proceeding with the calculation, it is necessary to analyze the expression: does it contain brackets and what actions are there in it. After that, proceed to the calculations in the following order:

1) actions enclosed in brackets;

2) multiplication and division;

3) addition and subtraction.

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