Establish the order of actions in the examples. The order of execution of actions in expressions without and with parentheses
We will look at three examples in this article:
1. Examples with parentheses (addition and subtraction actions)
2. Examples with parentheses (addition, subtraction, multiplication, division)
3. Examples with a lot of action
1 Examples with parentheses (addition and subtraction operations)
Let's look at three examples. In each of them, the order of actions is indicated by red numbers:
We see that the order of actions in each example will be different, although the numbers and signs are the same. This happens because there are parentheses in the second and third examples.
*This rule is for examples without multiplication and division. We will look at the rules for examples with parentheses involving the operations of multiplication and division in the second part of this article.
To avoid confusion in the example with parentheses, you can turn it into a regular example, without parentheses. To do this, write the result obtained in brackets above the brackets, then rewrite the entire example, writing this result instead of brackets, and then perform all the actions in order, from left to right:
In simple examples, you can perform all these operations in your mind. The main thing is to first perform the action in brackets and remember the result, and then count in order, from left to right.
And now - simulators!
1) Examples with brackets up to 20. Online simulator.
2) Examples with brackets up to 100. Online simulator.
3) Examples with brackets. Simulator No. 2
4) Insert the missing number - examples with brackets. Training apparatus
2 Examples with parentheses (addition, subtraction, multiplication, division)
Now let's look at examples in which, in addition to addition and subtraction, there is multiplication and division.
Let's look at examples without parentheses first:
There is one trick to avoid getting confused when solving examples of the order of actions. If there are no parentheses, then we perform the operations of multiplication and division, then we rewrite the example, writing down the results obtained instead of these actions. Then we perform addition and subtraction in order:
If the example contains parentheses, then first you need to get rid of the parentheses: rewrite the example, writing the result obtained in them instead of the parentheses. Then you need to mentally highlight the parts of the example, separated by the signs “+” and “-“, and count each part separately. Then perform addition and subtraction in order:
3 Examples with a lot of action
If there are many actions in the example, then it will be more convenient not to arrange the order of actions in the entire example, but to select blocks and solve each block separately. To do this, we find free signs “+” and “–” (free means not in brackets, shown in the figure with arrows).
These signs will divide our example into blocks:
When performing actions in each block, do not forget about the procedure given above in the article. Having solved each block, we perform the addition and subtraction operations in order.
Now let’s consolidate the solution to the examples on the order of actions on the simulators!
This lesson discusses in detail the procedure for performing arithmetic operations in expressions without parentheses and with brackets. Students are given the opportunity, while completing assignments, to determine whether the meaning of expressions depends on the order in which arithmetic operations are performed, to find out whether the order of arithmetic operations is different in expressions without parentheses and with parentheses, to practice applying the learned rule, to find and correct errors made when determining the order of actions.
In life, we constantly perform some kind of action: we walk, study, read, write, count, smile, quarrel and make peace. We perform these actions in different orders. Sometimes they can be swapped, sometimes not. For example, when getting ready for school in the morning, you can first do exercises, then make your bed, or vice versa. But you can’t go to school first and then put on clothes.
Is it necessary to do this in mathematics? arithmetic operations in a certain order?
Let's check
Let's compare the expressions:
8-3+4 and 8-3+4
We see that both expressions are exactly the same.
Let's perform actions in one expression from left to right, and in the other from right to left. You can use numbers to indicate the order of actions (Fig. 1).
Rice. 1. Procedure
In the first expression, we will first perform the subtraction operation and then add the number 4 to the result.
In the second expression, we first find the value of the sum, and then subtract the resulting result 7 from 8.
We see that the meanings of the expressions are different.
Let's conclude: The order in which arithmetic operations are performed cannot be changed.
Let's learn the rule for performing arithmetic operations in expressions without parentheses.
If an expression without parentheses includes only addition and subtraction or only multiplication and division, then the actions are performed in the order in which they are written.
Let's practice.
Consider the expression
This expression contains only addition and subtraction operations. These actions are called first stage actions.
We perform the actions from left to right in order (Fig. 2).
Rice. 2. Procedure
Consider the second expression
This expression contains only multiplication and division operations - These are the actions of the second stage.
We perform the actions from left to right in order (Fig. 3).
Rice. 3. Procedure
In what order are arithmetic operations performed if the expression contains not only addition and subtraction, but also multiplication and division?
If an expression without parentheses includes not only the operations of addition and subtraction, but also multiplication and division, or both of these operations, then first perform in order (from left to right) multiplication and division, and then addition and subtraction.
Let's look at the expression.
Let's think like this. This expression contains the operations of addition and subtraction, multiplication and division. We act according to the rule. First, we perform in order (from left to right) multiplication and division, and then addition and subtraction. Let's arrange the order of actions.
Let's calculate the value of the expression.
18:2-2*3+12:3=9-6+4=3+4=7
In what order are arithmetic operations performed if there are parentheses in an expression?
If an expression contains parentheses, then the value of the expressions in the parentheses is evaluated first.
Let's look at the expression.
30 + 6 * (13 - 9)
We see that in this expression there is an action in parentheses, which means we will perform this action first, then multiplication and addition in order. Let's arrange the order of actions.
30 + 6 * (13 - 9)
Let's calculate the value of the expression.
30+6*(13-9)=30+6*4=30+24=54
How should one reason to correctly establish the order of arithmetic operations in a numerical expression?
Before starting calculations, you need to look at the expression (find out whether it contains parentheses, what actions it contains) and only then perform the actions in the following order:
1. actions written in brackets;
2. multiplication and division;
3. addition and subtraction.
The diagram will help you remember this simple rule(Fig. 4).
Rice. 4. Procedure
Let's practice.
Let's consider the expressions, establish the order of actions and perform calculations.
43 - (20 - 7) +15
32 + 9 * (19 - 16)
We will act according to the rule. The expression 43 - (20 - 7) +15 contains operations in parentheses, as well as addition and subtraction operations. Let's establish a procedure. The first action is to perform the operation in parentheses, and then, in order from left to right, subtraction and addition.
43 - (20 - 7) +15 =43 - 13 +15 = 30 + 15 = 45
The expression 32 + 9 * (19 - 16) contains operations in parentheses, as well as multiplication and addition operations. According to the rule, we first perform the action in parentheses, then multiplication (we multiply the number 9 by the result obtained by subtraction) and addition.
32 + 9 * (19 - 16) =32 + 9 * 3 = 32 + 27 = 59
In the expression 2*9-18:3 there are no parentheses, but there are multiplication, division and subtraction operations. We act according to the rule. First, we perform multiplication and division from left to right, and then subtract the result obtained from division from the result obtained by multiplication. That is, the first action is multiplication, the second is division, and the third is subtraction.
2*9-18:3=18-6=12
Let's find out whether the order of actions in the following expressions is correctly defined.
37 + 9 - 6: 2 * 3 =
18: (11 - 5) + 47=
7 * 3 - (16 + 4)=
Let's think like this.
37 + 9 - 6: 2 * 3 =
There are no parentheses in this expression, which means that we first perform multiplication or division from left to right, then addition or subtraction. In this expression, the first action is division, the second is multiplication. The third action should be addition, the fourth - subtraction. Conclusion: the procedure is determined correctly.
Let's find the value of this expression.
37+9-6:2*3 =37+9-3*3=37+9-9=46-9=37
Let's continue to talk.
The second expression contains parentheses, which means that we first perform the action in parentheses, then from left to right multiplication or division, addition or subtraction. We check: the first action is in parentheses, the second is division, the third is addition. Conclusion: the procedure is defined incorrectly. Let's correct the errors and find the meaning of the expression.
18:(11-5)+47=18:6+47=3+47=50
This expression also contains parentheses, which means that we first perform the action in parentheses, then from left to right multiplication or division, addition or subtraction. Let's check: the first action is in parentheses, the second is multiplication, the third is subtraction. Conclusion: the procedure is defined incorrectly. Let's correct the errors and find the meaning of the expression.
7*3-(16+4)=7*3-20=21-20=1
Let's complete the task.
Let's arrange the order of actions in the expression using the learned rule (Fig. 5).
Rice. 5. Procedure
We don't see numerical values, so we won't be able to find the meaning of expressions, but we'll practice applying the rule we've learned.
We act according to the algorithm.
The first expression contains parentheses, which means the first action is in parentheses. Then from left to right multiplication and division, then from left to right subtraction and addition.
The second expression also contains parentheses, which means we perform the first action in parentheses. After that, from left to right, multiplication and division, after that, subtraction.
Let's check ourselves (Fig. 6).
Rice. 6. Procedure
Today in class we learned about the rule for the order of actions in expressions without and with brackets.
Bibliography
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 1. - M.: “Enlightenment”, 2012.
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 2. - M.: “Enlightenment”, 2012.
- M.I. Moro. Mathematics lessons: Methodological recommendations for teachers. 3rd grade. - M.: Education, 2012.
- Regulatory document. Monitoring and evaluation of learning outcomes. - M.: “Enlightenment”, 2011.
- "School of Russia": Programs for primary school. - M.: “Enlightenment”, 2011.
- S.I. Volkova. Mathematics: Test 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.
And the division of numbers is by actions of the second stage.
The order of actions when finding the values of expressions is determined by the following rules:
1. If there are no parentheses in the expression and it contains actions of only one stage, then they are performed in order from left to right.
2. If the expression contains actions of the first and second stages and there are no parentheses in it, then the actions of the second stage are performed first, then the actions of the first stage.
3. If there are parentheses in the expression, then perform the actions in the parentheses first (taking into account rules 1 and 2).
Example 1. Let's find the value of the expression
a) x + 20 = 37;
b) y + 37 = 20;
c) a - 37 = 20;
d) 20 - m = 37;
e) 37 - s = 20;
e) 20 + k = 0.
636. When subtracting which natural numbers maybe it will be 12? How many pairs of such numbers? Answer the same questions for multiplication and division.
637. Three numbers are given: the first is a three-digit number, the second is the quotient of a six-digit number divided by ten, and the third is 5921. Is it possible to indicate the largest and smallest of these numbers?
638. Simplify the expression:
a) 2a + 612 + 1a + 324;
b) 12у + 29у + 781 + 219;
639. Solve the equation:
a) 8x - 7x + 10 = 12;
b) 13y + 15y- 24 = 60;
c) Зz - 2z + 15 = 32;
d) 6t + 5t - 33 = 0;
e) (x + 59) : 42 = 86;
e) 528: k - 24 = 64;
g) p: 38 - 76 = 38;
h) 43m- 215 = 473;
i) 89n + 68 = 9057;
j) 5905 - 21 v = 316;
k) 34s - 68 = 68;
m) 54b - 28 = 26.
640. A livestock farm provides a weight gain of 750 g per animal per day. What gain does the complex receive in 30 days for 800 animals?
641. There are 130 liters of milk in two large and five small cans. How much milk does a small can contain if its capacity is four times less than the capacity of a larger one?
642. The dog saw its owner when it was 450 m away from him and ran towards him at a speed of 15 m/s. What will be the distance between the owner and the dog in 4 s; after 10 s; in t s?
643. Solve the problem using the equation:
1) Mikhail has 2 times more nuts than Nikolai, and Petya has 3 times more than Nikolai. How many nuts does each person have if everyone has 72 nuts?
2) Three girls collected 35 shells on the seashore. Galya found 4 times more than Masha, and Lena found 2 times more than Masha. How many shells did each girl find?
644. Write a program to evaluate the expression
8217 + 2138 (6906 - 6841) : 5 - 7064.
Write this program in diagram form. Find the meaning of the expression.
645. Write an expression using the following calculation program:
1. Multiply 271 by 49.
2. Divide 1001 by 13.
3. Multiply the result of command 2 by 24.
4. Add the results of commands 1 and 3.
Find the meaning of this expression.
646. Write an expression according to the diagram (Fig. 60). Write a program to calculate it and find its value.
647. Solve the equation:
a) Zx + bx + 96 = 1568;
b) 357z - 1492 - 1843 - 11 469;
c) 2y + 7y + 78 = 1581;
d) 256m - 147m - 1871 - 63,747;
e) 88 880: 110 + x = 809;
f) 6871 + p: 121 = 7000;
g) 3810 + 1206: y = 3877;
h) k + 12 705: 121 = 105.
648. Find the quotient:
a) 1,989,680: 187; c) 9 018 009: 1001;
b) 572 163: 709; d) 533,368,000: 83,600.
649. The motor ship traveled along the lake for 3 hours at a speed of 23 km/h, and then along the river for 4 hours. How many kilometers did the ship travel in these 7 hours if it moved along the river 3 km/h faster than along the lake?
650. Now the distance between the dog and the cat is 30 m. In how many seconds will the dog catch up with the cat if the dog’s speed is 10 m/s, and the cat’s is 7 m/s?
651. Find in the table (Fig. 61) all the numbers in order from 2 to 50. It is useful to perform this exercise several times; You can compete with a friend: who can find all the numbers faster?
N.Ya. VILENKIN, V. I. ZHOKHOV, A. S. CHESNOKOV, S. I. SHVARTSBURD, Mathematics grade 5, Textbook for educational institutions
Lesson plans for 5th grade mathematics download, textbooks and books for free, development of mathematics lessons online
Lesson content lesson notes supporting frame lesson presentation acceleration methods interactive technologies Practice tasks and exercises self-test workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photographs, pictures, graphics, tables, diagrams, humor, anecdotes, jokes, comics, parables, sayings, crosswords, quotes Add-ons abstracts articles tricks for the curious cribs textbooks basic and additional dictionary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in a textbook, elements of innovation in the lesson, replacing outdated knowledge with new ones Only for teachers perfect lessons calendar plan for the year guidelines discussion programs Integrated LessonsWhen we work with various expressions that include numbers, letters and variables, we have to perform a large number of arithmetic operations. When we do a conversion or calculate a value, it is very important to follow the correct order of these actions. In other words, arithmetic operations have their own special order of execution.
Yandex.RTB R-A-339285-1
In this article we will tell you which actions should be done first and which ones after. First, let's look at a few simple expressions that contain only variables or numeric values, as well as division, multiplication, subtraction and addition signs. Then let's take examples with parentheses and consider in what order they should be calculated. In the third part we will give the necessary order of transformations and calculations in those examples that include signs of roots, powers and other functions.
Definition 1In 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 ones numeric expressions, so that all calculations can be carried out in the mind. This way you can quickly remember the desired order and quickly check the results.
Example 1
Condition: calculate how much it will be 7 − 3 + 6 .
Solution
There are no parentheses in our expression, there is also no multiplication and division, so we perform all the actions in the specified order. First we subtract three from seven, then add six to the remainder and end up with ten. Here is a transcript of the entire solution:
7 − 3 + 6 = 4 + 6 = 10
Answer: 7 − 3 + 6 = 10 .
Example 2
Condition: in what order should the calculations be performed in the expression? 6:2 8:3?
Solution
To answer this question, let’s reread the rule for expressions without parentheses that we formulated earlier. We only have multiplication and division here, which means we keep the written order of calculations and count sequentially from left to right.
Answer: First we divide six by two, multiply the result by eight and divide the resulting number by three.
Example 3
Condition: calculate how much it will be 17 − 5 · 6: 3 − 2 + 4: 2.
Solution
First, let's determine the correct order of operations, since we have all the basic types of arithmetic operations here - addition, subtraction, multiplication, division. The first thing we need to do is divide and multiply. These actions do not have priority over each other, so we perform them in the written order from right to left. That is, 5 must be multiplied by 6 to get 30, then 30 divided by 3 to get 10. After that, divide 4 by 2, this is 2. Let's substitute the found values into the original expression:
17 − 5 6: 3 − 2 + 4: 2 = 17 − 10 − 2 + 2
There is no longer division or multiplication here, so we do the remaining calculations in order and get the answer:
17 − 10 − 2 + 2 = 7 − 2 + 2 = 5 + 2 = 7
Answer:17 − 5 6: 3 − 2 + 4: 2 = 7.
Until the order of performing actions is firmly memorized, you can put numbers above the signs of arithmetic operations indicating the order of calculation. For example, for the problem above we could write it like this:
If we have letter expressions, then we do the same with them: first we multiply and divide, then we add and subtract.
What are the first and second stage actions?
Sometimes in reference books all arithmetic operations are divided into actions of the first and second stages. Let us formulate the necessary definition.
The operations of the first stage include subtraction and addition, the second - multiplication and division.
Knowing these names, we can write the previously given rule regarding the order of actions as follows:
Definition 2
In an expression that does not contain parentheses, you must first perform the actions of the second stage in the direction from left to right, then the actions of the first stage (in the same direction).
Order of calculations in expressions with parentheses
The parentheses themselves are a sign that tells us the desired order of actions. In this case the right rule can be written like this:
Definition 3
If there are parentheses in the expression, then the first step is to perform the operation in them, after which we multiply and divide, and then add and subtract from left to right.
As for the parenthetical expression itself, it can be considered as an integral part of the main expression. When calculating the value of the expression in brackets, we maintain the same procedure known to us. Let's illustrate our idea with an example.
Example 4
Condition: calculate how much it will be 5 + (7 − 2 3) (6 − 4) : 2.
Solution
There are parentheses in this expression, so let's start with them. First of all, let's calculate how much 7 − 2 · 3 will be. Here we need to multiply 2 by 3 and subtract the result from 7:
7 − 2 3 = 7 − 6 = 1
We calculate the result in the second brackets. There we have only one action: 6 − 4 = 2 .
Now we need to substitute the resulting values into the original expression:
5 + (7 − 2 3) (6 − 4) : 2 = 5 + 1 2: 2
Let's start with multiplication and division, then perform subtraction and get:
5 + 1 2: 2 = 5 + 2: 2 = 5 + 1 = 6
This concludes the calculations.
Answer: 5 + (7 − 2 3) (6 − 4) : 2 = 6.
Don't be alarmed if our condition contains an expression in which some parentheses enclose others. We only need to apply the rule above consistently to all expressions in parentheses. Let's take this problem.
Example 5
Condition: calculate how much it will be 4 + (3 + 1 + 4 (2 + 3)).
Solution
We have parentheses within parentheses. We start with 3 + 1 + 4 · (2 + 3), namely 2 + 3. It will be 5. The value will need to be substituted into the expression and calculated that 3 + 1 + 4 · 5. We remember that we first need to multiply and then add: 3 + 1 + 4 5 = 3 + 1 + 20 = 24. Substituting the found values into the original expression, we calculate the answer: 4 + 24 = 28 .
Answer: 4 + (3 + 1 + 4 · (2 + 3)) = 28.
In other words, when calculating the value of an expression that includes parentheses within parentheses, we start with the inner parentheses and work our way to the outer ones.
Let's say we need to find how much (4 + (4 + (4 − 6: 2)) − 1) − 1 will be. We start with the expression in the inner brackets. Since 4 − 6: 2 = 4 − 3 = 1, the original expression can be written as (4 + (4 + 1) − 1) − 1. Looking again at the inner parentheses: 4 + 1 = 5. We have come to the expression (4 + 5 − 1) − 1 . We count 4 + 5 − 1 = 8 and as a result we get the difference 8 - 1, the result of which will be 7.
The order of calculation in expressions with powers, roots, logarithms and other functions
If our condition contains an expression with a degree, root, logarithm or trigonometric function(sine, cosine, tangent and cotangent) or other functions, then first of all we calculate the value of the function. After this, we act according to the rules specified in the previous paragraphs. In other words, functions are equal in importance to the expression enclosed in brackets.
Let's look at an example of such a calculation.
Example 6
Condition: find how much is (3 + 1) · 2 + 6 2: 3 − 7.
Solution
We have an expression with a degree, the value of which must be found first. We count: 6 2 = 36. Now let’s substitute the result into the expression, after which it will take the form (3 + 1) · 2 + 36: 3 − 7.
(3 + 1) 2 + 36: 3 − 7 = 4 2 + 36: 3 − 7 = 8 + 12 − 7 = 13
Answer: (3 + 1) 2 + 6 2: 3 − 7 = 13.
In a separate article devoted to calculating the values of expressions, we provide other, more complex examples calculations in the case of expressions with roots, degrees, etc. We recommend that you familiarize yourself with it.
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