We will look at three options for examples in this article:
1. Examples with brackets (addition and subtraction actions)
2. Examples with brackets (addition, subtraction, multiplication, division)
3. Examples with many actions
1 Examples with brackets (addition and subtraction actions)
Let's look at three examples. In each of them, the procedure is indicated by red numbers:
We see that the order of actions in each example will be different, although the numbers and signs are the same. This is because there are parentheses in the second and third examples.
* This rule is for non-multiplication and division examples. We'll cover rules for parenthetical examples involving multiplication and division in the second part of this article.
To avoid confusion in the parenthesized example, you can turn it into a regular example without parentheses. To do this, write the result obtained in parentheses above the parentheses, then rewrite the whole example, writing this result instead of parentheses, and then perform all the actions in order, from left to right:
In simple examples, all these operations can be performed in the mind. The main thing is to first perform the action in brackets and remember the result, and then count in order, from left to right.
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 brackets (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 how not to get confused when solving examples in the order of actions. If there are no brackets, then we perform the operations of multiplication and division, then we rewrite the example, writing down the results obtained instead of these actions. Then we add and subtract in order:
If the example contains parentheses, then first you need to get rid of the parentheses: rewrite the example by 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 add and subtract 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 whole example, but to select the blocks and solve each block separately. To do this, we find free signs "+" and "-" (free - it means not in brackets, shown by arrows in the figure).
These signs will divide our example into blocks:
When performing actions in each block, do not forget about the procedure described above in the article. Having solved each block, we perform addition and subtraction in order.
And now we fix the solution of examples on the order of actions on the simulators!
To correctly evaluate expressions in which you need to perform more than one operation, you need to know the order in which arithmetic operations are performed. We agreed to perform arithmetic operations in an expression without parentheses in the following order:
- If the expression contains exponentiation, then first this action is performed in the order, that is, from left to right.
- Then (if present in the expression) the operations of multiplication and division are performed in the order in which they appear.
- The last (if present in the expression) are addition and subtraction actions in the order of their sequence.
As an example, consider the following expression:
First you need to perform exponentiation (square 4 and cube 2):
3 16 - 8: 2 + 20
Then multiplication and division are performed (3 times 16 and 8 times 2):
And at the very end, subtraction and addition are performed (subtract 4 from 48 and add 20 to the result):
48 - 4 + 20 = 44 + 20 = 64
Actions of the first and second stage
Arithmetic operations are divided into first and second stage operations. Addition and subtraction are called first step actions, multiplication and division - second tier actions.
If the expression contains actions of only one step and there are no brackets in it, then the actions are performed in the order they follow from left to right.
Example 1.
15 + 17 - 20 + 8 - 12
Solution. This expression contains the actions of only one step - the first (addition and subtraction). It is necessary to determine the order of actions and carry them out.
Answer: 42.
If the expression contains the actions of both stages, then the actions of the second stage are performed first, in the order they follow (from left to right), and then the actions of the first stage.
Example. Calculate the value of the expression:
24: 3 + 5 2 - 17
Solution. This expression contains four actions: two of the first stage and two of the second. Let's determine the order of their execution: according to the rule, the first action will be division, the second is multiplication, the third is addition, and the fourth is subtraction.
Now let's get down to the calculation.
This lesson describes in detail the order of performing arithmetic operations in expressions without and with brackets. Students are given the opportunity, in the course of completing the assignments, to determine whether the value of expressions depends on the order of performing arithmetic operations, to find out whether the order of arithmetic operations in expressions without brackets and with brackets is different, to practice applying the learned rule, to find and correct mistakes made in determining the order of actions.
In life, we constantly perform any actions: we walk, study, read, write, count, smile, quarrel and make peace. We perform these actions in a different order. Sometimes they can be swapped and sometimes not. For example, getting ready for 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 your clothes.
And in mathematics, is it necessary to perform arithmetic operations in a certain order?
Let's check
Let's compare 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 another from right to left. Numbers can be used to indicate the order of actions (Fig. 1).
Rice. 1. Procedure
In the first expression, we'll first subtract and then add 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 values of the expressions are different.
Let's conclude: the order of performing arithmetic operations cannot be changed.
Let's learn the rule of performing arithmetic operations in expressions without brackets.
If an 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
In this expression, there are only addition and subtraction actions. 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 multiplication and division actions - these are the actions of the second stage.
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 an expression without brackets includes not only addition and subtraction, but also multiplication and division, or both of these actions, then first multiply and divide in order (from left to right), and then add and subtract.
Consider the 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 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 the expression?
If the expression contains parentheses, then the value of the expressions in parentheses is calculated first.
Consider the expression.
30 + 6 * (13 - 9)
We see that this expression contains an action in brackets, which means that we will perform this action first, then, in order, multiplication and addition. 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 in order to correctly establish the order of arithmetic operations in a numeric expression?
Before proceeding with the calculations, you need to consider the expression (find out if it contains brackets, 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 look at the expressions, set the order of actions, and perform the calculations.
43 - (20 - 7) +15
32 + 9 * (19 - 16)
We will act according to the rule. Expression 43 - (20 - 7) +15 contains operations in parentheses, as well as addition and subtraction operations. Let's establish the order of actions. The first action 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) contains actions in parentheses, as well as multiplication and addition actions. According to the rule, we first perform the action in parentheses, then multiply (the number 9 is multiplied by the result obtained by subtraction) and addition.
32 + 9 * (19 - 16) =32 + 9 * 3 = 32 + 27 = 59
There are no parentheses in the expression 2 * 9-18: 3, but there are operations of multiplication, division and subtraction. We act according to the rule. First, let's perform multiplication and division from left to right, and then subtract the result obtained from division from the result obtained by multiplying. 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 is defined correctly in the following expressions.
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 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 must be addition, the fourth is subtraction. Conclusion: the order of actions is defined correctly.
Let's find the value of this expression.
37+9-6:2*3 =37+9-3*3=37+9-9=46-9=37
We continue to reason.
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. Check: the first action is in brackets, the second is division, and the third is addition. Conclusion: the order of actions is defined incorrectly. Let's fix the errors, find the value 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. Check: the first action is in brackets, the second is multiplication, and the third is subtraction. Conclusion: the order of actions is defined incorrectly. Let's fix 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 learned rule (Fig. 5).
Rice. 5. Procedure
We do not see the numerical values, so we cannot 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, so the first action is in parentheses. Then multiplication and division from left to right, then subtraction and addition from left to right.
The second expression also contains parentheses, which means that the first action is performed 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 the lesson we got acquainted with the rule of the order of actions in expressions without brackets and with brackets.
Bibliography
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Education", 2012.
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Education", 2012.
- M.I. Moreau. Mathematics Lessons: Guidelines for Teachers. Grade 3. - M .: Education, 2012.
- Normative legal document. Monitoring and evaluation of learning outcomes. - M .: "Education", 2011.
- "School of Russia": Programs for elementary school. - M .: "Education", 2011.
- S.I. Volkova. Mathematics: Verification work. Grade 3. - 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 expressions.
2. Determine in what expression this order of performing actions:
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.
Task 192.
Complete the assignments verbally.
- 1) Find the sum of the numbers 5 and 2. Subtract this sum from the number 10.
- 2) Add the difference between the numbers 9 and 3 to the number 8.
Solution:
- 1) 10 - (5 + 2) = 3
- 2) 8 + (9 - 3) = 14
Task 193.
There were 15 m of fabric in a roll. The first customer bought 5 m of fabric, and the second bought 3 m. How many meters of fabric are left on the roll?
To find out how many meters of fabric remained on the roll, the seller did this: he calculated how many meters of fabric he sold, and then subtracted the resulting number from 15.
The parentheses indicate that first it is clear to find the amount, and then perform the subtraction action.
Task 194.
Read and calculate.
Subtract the sum of the numbers 7 and 2 from the number 12.
Add the difference between 13 and 6 to the number 8.
Solution:
- 1) 12 - (7 + 2) = 3
- 2) 8 + (13 - 6) = 15
Task 195.
There were 12 cars in the parking lot. First, 4 cars drove off, and then - another 3. How many cars are left in the parking lot?
Solution:
- 1) 12 - (4 + 3) = 5
- Answer: 5 cars.
Task 196.
One protein has 9 nuts and the other has the same number. How many nuts do proteins have?
Solution:
- 1) 9 + 9 = 18
- Answer: 18 nuts.
Task 197.
Read and calculate.
- 1) From the number 14, subtract the difference between the numbers 7 and 2.
- 2) Add to the number 8 the sum of the numbers 3 and 6.
Solution:
- 1) 14 - (7 - 2) = 9
- 2) 8 + (3 + 6) = 17
Task 198.
There were 13 trucks in the parking lot, and 8 fewer cars. 6 more cars arrived. How many cars are in the parking lot?
Solution:
- 1) (13 - 8) + 6 = 11
- Answer: 11 cars.
Task 199.
Complete and solve the problem.
One classroom has 7 computers, and the other has 2 computers ....
Solution:
One classroom has 7 computers, and the other has 2 fewer computers. How many computers are in 2 classes together.
- 1) 7 - 2 = 5
- 2) 7 + 5 = 12
- Expression: (7 - 2) + 7 = 12
- Answer: 12 computers.
Task 200.
Solve examples.
Solution:
Solution:
Task 202.
From each addition example, compose two subtraction examples.
| 9 + 7 = 16 | 14 - 6 = 8 |
Solution:
Task 204.
Solution:
- 1) Add 9 and 7 equals 16. 9 plus 7 equals 16. 9 increase by 7 equals 16. The sum of nine and seven equals sixteen.
- 2) 14 subtract 6 equals 8. 14 minus 6 equals 8. 14 subtract 6 equals 8. The difference between fourteen and six equals eight.
Task 205.
In the morning from the cow milked 9 liters, | and in the evening - 1 liter less. | 3 liters of milk from the evening milk left, | and the rest was sold. How many liters of milk from the evening milk yield have you sold?
Read the entire problem. Think about what it is about.
Read the problem in parts, into which it is divided by lines.
Solve the problem.
Solution plan
- 1) How many liters of milk did you drink in the evening?
- 2) How many liters of milk from the evening milk yield have you sold?
Solution:
- 1) 9 - 1 = 8
- 2) 8 - 3 = 5
- Expression: (9 - 1) - 3 = 5
- Answer: 5 liters.
Task 206.
On Saturday, father and son cut 4 trees together. On Sunday, the father pruned 3 trees and the son pruned the same number of trees. How many trees did they cut in 2 days?
Solution:
- 1) 3 + 3 = 6
- 2) 4 + 6 = 10
- Expression: 4 + 3 + 3 = 10
- Answer: 10 trees.
Task 207.
Solve examples.
Solution:
| 14 - 6 - 6 = 2 | 7 + 5 + 1 = 13 | 16 - 8 + 1 = 9 |
| 14 - (6 - 6) = 14 | 7 + (5 + 1) = 13 | 16 - (8 + 1) = 7 |
Task 208.
Create a drawing problem and solve it. 
Solution:
There were 12 apples under the tree. One hedgehog took 4 apples, and the other took another 3. How many apples are left under the tree?
- 1) 4 + 3 = 7
- 2) 12 - 7 = 5
- Expression: 12 - (4 + 3) = 5
- Answer: 5 apples.
Today we will talk about order of execution mathematical action... What are the first steps to take? Addition and subtraction, or multiplication and division. Strange, but our children have problems solving seemingly elementary expressions.
So, remember that the expressions in parentheses are evaluated first.
38 – (10 + 6) = 22 ;
Procedure for performing actions:
1) in brackets: 10 + 6 = 16;
2) subtraction: 38 - 16 = 22.
If an expression without brackets includes only addition and subtraction, or only multiplication and division, then the actions are performed in order from left to right.
10 ÷ 2 × 4 = 20;
Procedure for performing actions:
1) from left to right, first division: 10 ÷ 2 = 5;
2) multiplication: 5 × 4 = 20;
10 + 4 - 3 = 11, i.e .:
1) 10 + 4 = 14 ;
2) 14 – 3 = 11 .
If an expression without brackets contains not only addition and subtraction, but also multiplication or division, then the actions are performed in order from left to right, but multiplication and division have the advantage, they are performed first of all, followed by addition and subtraction.
18 ÷ 2 - 2 × 3 + 12 ÷ 3 = 7
Procedure for performing actions:
1) 18 ÷ 2 = 9;
2) 2 × 3 = 6;
3) 12 ÷ 3 = 4;
4) 9 - 6 = 3; those. from left to right - the result of the first action minus the result of the second;
5) 3 + 4 = 7; those. the result of the fourth action plus the result of the third;
If the expression contains parentheses, then the expressions in parentheses are executed first, then multiplication and division, and only then addition and subtraction.
30 + 6 × (13 - 9) = 54, i.e .:
1) expression in brackets: 13 - 9 = 4;
2) multiplication: 6 × 4 = 24;
3) addition: 30 + 24 = 54;
So, let's summarize. Before proceeding with the calculation, you need to analyze the expression: does it contain parentheses and what actions it contains. After that, proceed with the calculations in the following order:
1) actions enclosed in brackets;
2) multiplication and division;
3) addition and subtraction.
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