The sequence of performing mathematical operations. Summary of the lesson ""The order of execution of actions in expressions without brackets and with brackets."

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.

The order of actions - Mathematics Grade 3 (Moro)

Short description:

In life, you constantly perform various actions: get up, wash your face, do exercises, have breakfast, go to school. Do you think this procedure can be changed? For example, have breakfast, and then wash. Probably you can. It may not be very convenient to have breakfast unwashed, but nothing terrible will happen because of this. And in mathematics, is it possible to change the order of actions at will? No, mathematics is an exact science, so even the slightest change in the order of operations will cause the answer of a numerical expression to become incorrect. In the second grade, you already got acquainted with some rules of the order of actions. So, you probably remember that parentheses govern the order in performing actions. They indicate that actions must be performed first. What other rules of procedure are there? Is the order of operations in expressions with brackets and without brackets different? You will find answers to these questions in the 3rd grade mathematics textbook when studying the topic "Order of actions". You must definitely practice applying the learned rules, and if necessary, find and correct errors in establishing the order of actions in numerical expressions. Please remember that order is important in any business, but in mathematics it has a special meaning!

If we compare the functions of addition and subtraction with multiplication and division, then multiplication and division are always calculated first.

In the example, two functions such as addition and subtraction, as well as multiplication and division, are equivalent to each other. The order of execution is determined in turn order from left to right.

It should be remembered that the actions taken in parentheses have special precedence in the example. Thus, even if there is multiplication outside the brackets, and addition in brackets, you should first add, and only then multiply.

To understand this topic, you can consider all cases in turn.

Immediately take into account that our expressions do not have brackets.

So, if in the example the first action is multiplication, and the second is division, then we perform the multiplication first.

If in the example the first action is division, and the second is multiplication, then we do division first.

In such examples, actions are performed in order from left to right, regardless of which numbers are used.

If, in addition to multiplication and division, there are addition and subtraction in the examples, then multiplication and division are done first, and then addition and subtraction.

In the case of addition and subtraction, it also does not matter which of these operations is done first. The order is from left to right.

Let's consider different options:

In this example, the first action that needs to be performed is multiplication, and then addition.

In this case, you first multiply the values, then divide, and only then add.

In this case, you must first do all the operations in the brackets, and then only do the multiplication and division.

And so it must be remembered that in any formula, operations are first performed as multiplication and division, and then only subtraction and addition.

Also, with the numbers that are in brackets, you need to count them in brackets, and only then do various manipulations, remembering the sequence described above.

The first will be the following actions: multiplication and division.

Only then are addition and subtraction performed.

However, if there is a bracket, then the actions that are in them will be performed first. Even if it's addition and subtraction.

For example:

In this example, first we perform the multiplication, then 4 by 5, then add 4 to 20. We get 24.

But if it is like this: (4 + 5) * 4, then first we perform the addition, we get 9. Then we multiply 9 by 4. We get 36.

If all 4 actions are present in the example, then multiplication and division come first, and then addition and subtraction.

Or in the example of 3 different actions, then the first will be either multiplication (or division), and then either addition (or subtraction).

When there are NO BRACKETS.

Example: 4-2*5:10+8=11,

1 action 2*5 (10);

act 2 10:10 (1);

3 act 4-1 (3);

4 act 3+8 (11).

All 4 actions can be divided into two main groups, in one - addition and subtraction, in the other - multiplication and division. The first action will be the one that is the first in a row in the example, that is, the leftmost one.

Example: 60-7+9=62, first you need 60-7, then what happens (53) +9;

Example: 5*8:2=20, first you need 5*8, then what you get (40) :2.

When there are BRACKETS in the example, then the actions that are in the bracket are performed first (according to the above rules), and then the rest as usual.

Example: 2+(9-8)*10:2=7.

1 act 9-8 (1);

2 action 1*10 (10);

Act 3 10:2(5);

4 act 2+5 (7).

It depends on how the expression is written, consider the simplest numeric expression:

18 - 6:3 + 10x2 =

First, we perform operations with division and multiplication, then in turn, from left to right, with subtraction and addition: 18-2 + 20 \u003d 36

If it's a parenthesized expression, then do the parentheses, then multiply or divide, and finally add/subtract, like this:

(18-6): 3 + 10 x 2 = 12:3 + 20 = 4+20=24

Sun is correct: first perform multiplication and division, then addition and subtraction.

If there are no brackets in the example, then multiplication and division in order are performed first, and then addition and subtraction, the same in order.

If the example contains only multiplication and division, then the actions will be performed in order.

If the example contains only addition and subtraction, then the actions will also be performed in order.

First of all, actions in brackets are performed according to the same rules, that is, first multiplication and division, and only then addition and subtraction.

22-(11+3x2)+14=19

The order of performing arithmetic operations is strictly prescribed so that there are no discrepancies when performing the same type of calculations by different people. First of all, multiplication and division are performed, then addition and subtraction, if actions of the same order go one after the other, then they are performed in turn from left to right.

If brackets are used when writing a mathematical expression, then first of all, you should perform the actions indicated in brackets. Parentheses help to change the order, if necessary, first perform addition or subtraction, and only after multiplication and division.

Any brackets can be opened and then the execution order will again be correct:

6*(45+15) = 6*45 +6*15

Better with examples:

• 1+2*3/4-5=?

In this case, we perform the multiplication first, since it is to the left of the division. Then division. Then addition, because of the more left-hand location, and finally subtraction.

• 1*3/(2+4)?

first we do the calculation in brackets, then the multiplication and division.

• 1+2*(3-1*5)=?

First, we do the actions in brackets: multiplication, then subtraction. After that comes the multiplication outside the brackets and the addition at the end.

Multiplication and division come first. If there are brackets in the example, then the action in brackets is considered at the beginning. Whatever the sign is!

Here you need to remember a few basic rules:

1. If there are no brackets in the example and there are operations - only addition and subtraction, or only multiplication and division - in this case, all actions are carried out in order from left to right.

For example, 5 + 8-5 = 8 (we do everything in order - add 8 to 5, and then subtract 5)

1. If the example contains mixed operations - and addition, and subtraction, and multiplication, and division, then first of all we perform the operations of multiplication and division, and then only addition or subtraction.

For example, 5+8*3=29 (first multiply 8 by 3 and then add 5)

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

For example, 3*(5+8)=39 (first 5+8 and then multiply by 3)

And when 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. Next, we will explain what order of execution of actions should be followed in expressions with brackets. Finally, consider the sequence in which actions are performed in expressions containing powers, roots, and other functions.

Page navigation.

First multiplication and division, then addition and subtraction

The school provides 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,
• where 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 it is customary for us to keep records from left to right. And the fact that multiplication and division is performed before addition and subtraction is explained by the meaning that these actions carry in themselves.

Let's look at a few examples of the application of this rule. For examples, we will take the simplest numerical expressions so as not to be distracted by calculations, but to focus on the order in which actions are performed.

Example.

Follow steps 7−3+6 .

Solution.

The original expression does not contain parentheses, nor does it contain multiplication and division. Therefore, we should perform all 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 4, we get 10.

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

Answer:

7−3+6=10 .

Example.

Indicate the order in which actions are performed in the expression 6:2·8:3 .

Solution.

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

Answer:

First 6 divided by 2, this quotient is multiplied by 8, finally, the result is divided by 3.

Example.

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

Solution.

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 instead of 5 6:3 in the original expression, and the value 2 instead of 4:2, we have 17−5 6:3−2+4:2=17−10−2+2.

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

Answer:

17−5 6:3−2+4:2=7 .

At first, in order not to confuse the order of performing actions when calculating the value of an expression, it is convenient to place numbers above the signs of actions corresponding 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 literal expressions.

Steps 1 and 2

In some textbooks on mathematics, there is a division of arithmetic operations into operations of the first and second steps. Let's deal with this.

Definition.

First step actions are called addition and subtraction, and multiplication and division are called second step actions.

In these terms, the rule from the previous paragraph, which determines the order in which actions are performed, will be written as follows: if the expression does not contain brackets, 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 execution of arithmetic operations in expressions with brackets

Expressions often contain parentheses to indicate the order in which the actions are to be performed. In this case a rule that specifies the order in which actions are performed in expressions with brackets, 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, expressions in brackets are considered as components of the original expression, and the order of actions already known to us is preserved in them. Consider the solutions of examples for greater clarity.

Example.

Perform the given steps 5+(7−2 3) (6−4):2 .

Solution.

The expression contains brackets, so let's first perform the operations in the expressions enclosed in these brackets. Let's start with the expression 7−2 3 . In it, you must first perform the multiplication, and only then the subtraction, we have 7−2 3=7−6=1 . We pass 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, first we perform multiplication and division from left to right, then subtraction, we get 5+1 2:2=5+2:2=5+1=6 . On this, all actions are completed, we adhered to the following order of their execution: 5+(7−2 3) (6−4):2 .

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

Answer:

5+(7−2 3)(6−4):2=6 .

It happens that an expression contains brackets within brackets. You should not be afraid of this, you just need to consistently apply the voiced rule for performing actions in expressions with brackets. Let's show an example solution.

Example.

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

Solution.

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 first perform actions in them. 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 it remains only to complete the actions: 4+24=28 .

Answer:

4+(3+1+4 (2+3))=28 .

In general, when parentheses within parentheses are present in an expression, it is often convenient to start with the inner parentheses and work your way to the outer ones.

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

The video lesson "Order of actions" explains in detail an important topic of mathematics - the sequence of performing arithmetic operations when solving an expression. During the video lesson, it is considered what priority various mathematical operations have, how it is used in the calculation of expressions, examples are given for mastering the material, the knowledge gained is summarized in solving tasks, where all the considered operations are available. With the help of a video lesson, the teacher has the opportunity to quickly achieve the goals of the lesson, increase its effectiveness. The video can be used as a visual material accompanying the teacher's explanation, as well as an independent part of the lesson.

The visual material uses techniques that help to better understand the topic, as well as remember important rules. With the help of color and different spelling, the features and properties of operations are highlighted, the features of solving examples are noted. Animation effects help to present a consistent learning material, as well as draw students' attention to important points. The video is voiced, therefore it is supplemented with teacher's comments that help the student understand and remember the topic.

The video tutorial starts by introducing the topic. Then it is noted that multiplication, subtraction are operations of the first stage, the 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 colored print. Then the rules that make up the order in which operations are performed are presented. The first order rule is displayed, which indicates that if there are no brackets in the expression, if there are actions of one stage, these actions must be performed in order. The second rule of order states that if there are actions of both stages and there are no brackets, the operations of the second stage are performed first, then the operations of the first stage are performed. The third rule establishes the order in which operations are performed for expressions that include parentheses. It is noted that in this case operations in parentheses are performed first. The wording of the rules is highlighted in color and recommended for memorization.

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

In the second example 780:39·212:156·13 it is required to evaluate the expression by performing actions according to the order. It is noted that this expression contains only operations of the second stage, without brackets. In this example, all actions are performed strictly from left to right. Below, the actions are painted in turn, gradually approaching the answer. The result of the calculation is the number 520.

In the third example, the solution of the example is considered, in which there are operations of both stages. It is noted that in this expression there are no brackets, but there are actions of both steps. According to the order of operations, operations of the second stage are performed, after that - operations of the first stage. Below, the solution is described by actions, in which three operations are performed first - multiplication, division, one more division. Then, with the found values ​​of the product and quotients, operations of the first stage are performed. During the solution, curly brackets combine the actions of each step for clarity.

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

The following is a note on when you can not write parentheses when solving expressions. It is noted that this is possible only in the case when the removal of parentheses does not change the order of operations. An example is the expression with brackets (53-12)+14, which contains only the operations of the first stage. By rewriting 53-12+14 with the brackets removed, it can be noted that the search order for the value will not change - first subtract 53-12=41, and then add 41+14=55. It is noted below that it is possible to change the order of operations when finding a solution to an expression using the properties of operations.

At the end of the video lesson, the studied material is summarized in the conclusion that each expression that needs to be solved defines a specific program for calculation, consisting of commands. An example of such a program is presented when describing the solution of a complex example, which is a quotient of (814+36 27) and (101-2052:38). The specified program contains the following steps: 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 step four.

At the end of the video lesson there is a list of questions that students are asked to answer. Among them are the ability to distinguish between the actions of the first and second stages, questions about the order in which actions are performed in expressions with actions of the same stage and different stages, and the order in which actions are performed when there are brackets in the expression.

The video lesson "Procedure for performing actions" is recommended to be used in a traditional school lesson to increase the effectiveness of the lesson. Also visual material will be useful for distance learning. If the student needs an additional lesson to master the topic or he is studying it on his own, the video can be recommended for self-study.