Rule for solving examples on actions with brackets. Educational and methodological material in mathematics (grade 3) on the topic: Examples of the order of actions

October 24th, 2017 admin

Lopatko Irina Georgievna

Target: formation of knowledge about the order of execution arithmetic operations V numerical expressions without brackets and with brackets, consisting of 2-3 actions.

Tasks:

Educational: to develop in students the ability to use the rules of the order of actions when calculating specific expressions, the ability to apply an algorithm of actions.

Developmental: develop skills of working in pairs, mental activity of students, the ability to reason, compare and contrast, calculation skills and mathematical speech.

Educational: cultivate interest in the subject, tolerant attitude towards each other, mutual cooperation.

Type: learning new material

Equipment: presentation, visuals, handouts, cards, textbook.

Methods: verbal, visual and figurative.

DURING THE CLASSES

Organizing time

Greetings.

We came here to study

Don't be lazy, but work hard.

We work diligently

Let's listen carefully.

Markushevich said great words: “Whoever studies mathematics from childhood develops attention, trains his brain, his will, cultivates perseverance and perseverance in achieving goals.” Welcome to math lesson!

Updating knowledge

The subject of mathematics is so serious that no opportunity should be missed to make it more entertaining.(B. Pascal)

I suggest you complete logical tasks. You are ready?

Which two numbers, when multiplied, give the same result as when added? (2 and 2)

From under the fence you can see 6 pairs of horse legs. How many of these animals are there in the yard? (3)

A rooster standing on one leg weighs 5 kg. How much will he weigh standing on two legs? (5kg)

There are 10 fingers on the hands. How many fingers are there on 6 hands? (thirty)

The parents have 6 sons. Everyone has a sister. How many children are there in the family? (7)

How many tails do seven cats have?

How many noses do two dogs have?

How many ears do 5 babies have?

Guys, this is exactly the kind of work I expected from you: you were active, attentive, and smart.

Assessment: verbal.

Verbal counting

BOX OF KNOWLEDGE

Product of numbers 2 * 3, 4 * 2;

Partial numbers 15: 3, 10:2;

Sum of numbers 100 + 20, 130 + 6, 650 + 4;

The difference between numbers is 180 – 10, 90 – 5, 340 – 30.

Components of multiplication, division, addition, subtraction.

Assessment: students independently evaluate each other

Communicating the topic and purpose of the lesson

“To digest knowledge, you need to absorb it with appetite.”(A. Franz)

Are you ready to absorb knowledge with appetite?

Guys, Masha and Misha were offered such a chain

24 + 40: 8 – 4=

Masha decided it like this:

24 + 40: 8 – 4= 25 correct? Children's answers.

And Misha decided like this:

24 + 40: 8 – 4= 4 correct? Children's answers.

What surprised you? It seems that both Masha and Misha decided correctly. Then why do they have different answers?

They believed in in different order, have not agreed on the order in which they will count.

What does the calculation result depend on? From order.

What do you see in these expressions? Numbers, signs.

What are signs called in mathematics? Actions.

What order did the guys not agree on? About the procedure.

What will we study in class? What is the topic of the lesson?

We will study the order of arithmetic operations in expressions.

Why do we need to know the procedure? Perform calculations correctly in long expressions

"Basket of Knowledge". (The basket hangs on the board)

Students name associations related to the topic.

Learning new material

Guys, please listen to what the French mathematician D. Poya said: “The best way to study something is to discover it for yourself.” Are you ready for discoveries?

180 – (9 + 2) =

Read the expressions. Compare them.

How are they similar? 2 actions, same numbers

What is the difference? Brackets, different actions

Rule 1.

Read the rule on the slide. Children read the rule aloud.

In expressions without parentheses containing only addition and subtraction or multiplication and division, operations are performed in the order they are written: from left to right.

What actions are we talking about here? +, — or : , ·

From these expressions, find only those that correspond to rule 1. Write them down in your notebook.

Calculate the values of expressions.

Examination.

180 – 9 + 2 = 173

Rule 2.

Read the rule on the slide.

Children read the rule aloud.

In expressions without parentheses, multiplication or division are performed first, in order from left to right, and then addition or subtraction.

:, · and +, — (together)

Are there parentheses? No.

What actions will we perform first? ·, : from left to right

What actions will we take next? +, — left, right

Find their meanings.

Examination.

180 – 9 * 2 = 162

Rule 3

In expressions with parentheses, first evaluate the value of the expressions in parentheses, thenmultiplication or division are performed in order from left to right, and then addition or subtraction.

What arithmetic operations are indicated here?

:, · and +, — (together)

Are there parentheses? Yes.

What actions will we perform first? In brackets

What actions will we take next? ·, : from left to right

And then? +, — left, right

Write down expressions that relate to the second rule.

Find their meanings.

Examination.

180: (9 * 2) = 10

180 – (9 + 2) = 169

Once again, we all say the rule together.

PHYSMINUTE

Consolidation

“Much of mathematics does not remain in the memory, but when you understand it, then it is easy to remember what you have forgotten on occasion.”, said M.V. Ostrogradsky. Now we will remember what we just learned and apply new knowledge in practice .

Page 52 No. 2

(52 – 48) * 4 =

Page 52 No. 6 (1)

The students collected 700 kg of vegetables in the greenhouse: 340 kg of cucumbers, 150 kg of tomatoes, and the rest - peppers. How many kilograms of peppers did the students collect?

What are they talking about? What is known? What do you need to find?

Let's try to solve this problem with an expression!

700 – (340 + 150) = 210 (kg)

Answer: The students collected 210 kg of pepper.

Work in pairs.

Cards with the task are given.

5 + 5 + 5 5 = 35

(5+5) : 5 5 = 10

Grading:

speed – 1 b
correctness - 2 b
logic - 2 b

Homework

Page 52 No. 6 (2) solve the problem, write the solution in the form of an expression.

Result, reflection

Bloom's Cube

Name it topic of our lesson?

Explain the order of execution of actions in expressions with brackets.

Why Is it important to study this topic?

Continue first rule.

Come up with it algorithm for performing actions in expressions with brackets.

“If you want to participate in great life, then fill your head with mathematics while you have the opportunity. She will then be of great help to you in all your work.”(M.I. Kalinin)

Thanks for your work in class!!!

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When we work with various expressions that include numbers, letters and variables, we have to perform a large number of arithmetic operations. When we make 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 literal 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.

If you notice an error in the text, please highlight it and press Ctrl+Enter

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

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

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

Let's check

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

We see that both expressions are exactly the same.

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

Rice. 1. Procedure

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

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

We see that the meanings of the expressions are different.

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

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

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

Let's practice.

Consider the expression

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

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

Rice. 2. Procedure

Consider the second expression

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

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

Rice. 3. Procedure

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

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

Let's look at the expression.

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

Let's calculate the value of the expression.

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

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

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

Let's look at the expression.

30 + 6 * (13 - 9)

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

30 + 6 * (13 - 9)

Let's calculate the value of the expression.

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

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

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

1. actions written in brackets;

2. multiplication and division;

3. addition and subtraction.

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

Rice. 4. Procedure

Let's practice.

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

43 - (20 - 7) +15

32 + 9 * (19 - 16)

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

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

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

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

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

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

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

37 + 9 - 6: 2 * 3 =

18: (11 - 5) + 47=

7 * 3 - (16 + 4)=

Let's think like this.

37 + 9 - 6: 2 * 3 =

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

Let's find the value of this expression.

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

Let's continue to talk.

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

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

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

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

Let's complete the task.

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

Rice. 5. Procedure

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

We act according to the algorithm.

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

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

Let's check ourselves (Fig. 6).

Rice. 6. Procedure

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

Bibliography

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

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

Homework

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

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

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

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

1. multiplication; 2. addition; 3. subtraction

1. addition; 2. subtraction; 3. addition

1. multiplication; 2. division; 3. addition

Find the meaning of these expressions.

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 the cases in which you may 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.

Rules for the order of performing actions in complex expressions are studied in 2nd grade, but children practically use some of them in 1st grade.

First, we consider the rule about the order of operations in expressions without parentheses, when numbers are performed either only addition and subtraction, or only multiplication and division. The need to introduce expressions containing two or more arithmetic operations of the same level arises when students become familiar with the computational techniques of addition and subtraction within 10, namely:

Similarly: 6 - 1 - 1, 6 - 2 - 1, 6 - 2 - 2.

Since to find the meanings of these expressions, schoolchildren turn to objective actions that are performed in a certain order, they easily learn the fact that arithmetic operations (addition and subtraction) that take place in expressions are performed sequentially from left to right.

Students first encounter number expressions containing addition and subtraction operations and parentheses in the topic "Addition and Subtraction within 10." When children encounter such expressions in 1st grade, for example: 7 - 2 + 4, 9 - 3 - 1, 4 +3 - 2; in 2nd grade, for example: 70 - 36 +10, 80 - 10 - 15, 32+18 - 17; 4*10:5, 60:10*3, 36:9*3, the teacher shows how to read and write such expressions and how to find their meaning (for example, 4*10:5 read: 4 multiply by 10 and divide the resulting result at 5). By the time they study the topic “Order of Actions” in 2nd grade, students are able to find the meanings of expressions of this type. The goal of the work at this stage is based on practical skills students, draw their attention to the order of performing actions in such expressions and formulate the corresponding rule. Students independently solve examples selected by the teacher and explain in what order they performed them; actions in each example. Then they formulate the conclusion themselves or read from a textbook: if in an expression without parentheses only the actions of addition and subtraction (or only the actions of multiplication and division) are indicated, then they are performed in the order in which they are written (i.e., from left to right).

Despite the fact that in expressions of the form a+b+c, a+(b+c) and (a+b)+c the presence of parentheses does not affect the order of actions due to the associative law of addition, at this stage it is more advisable to orient students to that the action in parentheses is performed first. This is due to the fact that for expressions of the form a - (b + c) and a - (b - c) such a generalization is unacceptable and it will be quite difficult for students at the initial stage to navigate the assignment of brackets for various numerical expressions. The use of parentheses in numerical expressions containing addition and subtraction operations is further developed, which is associated with the study of such rules as adding a sum to a number, a number to a sum, subtracting a sum from a number and a number from a sum. But when first introducing parentheses, it is important to direct students to do the action in the parentheses first.

The teacher draws the children's attention to how important it is to follow this rule when making calculations, otherwise you may get an incorrect equality. For example, students explain how the meanings of the expressions are obtained: 70 - 36 +10 = 24, 60:10 - 3 = 2, why they are incorrect, what meanings these expressions actually have. Similarly, they study the order of actions in expressions with brackets of the form: 65 - (26 - 14), 50: (30 - 20), 90: (2 * 5). Students are also familiar with such expressions and can read, write and calculate their meaning. Having explained the order of actions in several such expressions, children formulate a conclusion: in expressions with brackets, the first action is performed on the numbers written in brackets. Looking at these expressions, it is not difficult to show that the actions in them are not performed in the order in which they are written; to show a different order of their execution, and parentheses are used.

The following introduces the rule for the order of execution of actions in expressions without parentheses, when they contain actions of the first and second stages. Since the rules of procedure are accepted by agreement, the teacher communicates them to the children or the students learn them from the textbook. To ensure that students understand the rules introduced, along with training exercises include solutions to examples with an explanation of the order of their actions. Exercises in explaining errors in the order of actions are also effective. For example, from the given pairs of examples, it is proposed to write down only those where the calculations were performed according to the rules of the order of actions:

After explaining the errors, you can give a task: using parentheses, change the order of actions so that the expression has the specified value. For example, in order for the first of the given expressions to have a value equal to 10, you need to write it like this: (20+30):5=10.

Exercises on calculating the value of an expression are especially useful when the student has to apply all the rules he has learned. For example, the expression 36:6+3*2 is written on the board or in notebooks. Students calculate its value. Then, according to the teacher’s instructions, the children use parentheses to change the order of actions in the expression:

36:6+3-2
36:(6+3-2)
36:(6+3)-2
(36:6+3)-2

An interesting, but more difficult, exercise is the reverse exercise: placing parentheses so that the expression has the given value:

72-24:6+2=66
72-24:6+2=6
72-24:6+2=10
72-24:6+2=69

Also interesting are the following exercises:

1. Arrange the brackets so that the equalities are true:
25-17:4=2 3*6-4=6
24:8-2=4
2. Place “+” or “-” signs instead of asterisks so that you get the correct equalities:
38*3*7=34
38*3*7=28
38*3*7=42
38*3*7=48
3. Place arithmetic signs instead of asterisks so that the equalities are true:
12*6*2=4
12*6*2=70
12*6*2=24
12*6*2=9
12*6*2=0

By performing such exercises, students become convinced that the meaning of an expression can change if the order of actions is changed.

To master the rules of the order of actions, it is necessary in grades 3 and 4 to include increasingly complex expressions, when calculating the values of which the student would apply not one, but two or three rules of the order of actions each time, for example:

90*8- (240+170)+190,
469148-148*9+(30 100 - 26909).

In this case, the numbers should be selected so that they allow actions to be performed in any order, which creates conditions for the conscious application of the learned rules.