The order of solving mathematical operations. Rules for solving examples on actions with brackets

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 what natural numbers can you get 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 general education institutions

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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 signs. Next, we will explain what order of actions should be followed in expressions with brackets. Finally, let's look at the order 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 gives 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,
Moreover, 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 are performed before addition and subtraction is explained by the meaning that these actions carry.

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

Example.

Follow steps 7−3+6.

Solution.

The original expression does not contain parentheses, and it does not contain multiplication or 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 of 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 of actions in the expression 6:2·8:3.

Solution.

To answer the question of the problem, let's turn to the rule indicating the order of execution of actions in expressions without parentheses. 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:

At first We divide 6 by 2, multiply this quotient by 8, and finally divide the result 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 into the original expression instead of 5·6:3, and instead of 4:2 - the value 2, we have 17−5·6:3−2+4:2=17−10−2+2.

The resulting expression no longer contains multiplication and division, 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 in which actions are performed when calculating the value of an expression, it is convenient to place numbers above the action signs that correspond 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 letter expressions.

Actions of the first and second stages

In some mathematics textbooks there is a division of arithmetic operations into operations of the first and second stages. Let's figure this out.

Definition.

Actions of the first stage addition and subtraction are called, and multiplication and division are called second stage actions.

In these terms, the rule from the previous paragraph, which determines the order of execution of actions, will be written as follows: if the expression does not contain parentheses, then in order from left to right, first the actions of the second stage (multiplication and division) are performed, then the actions of the first stage (addition and subtraction).

Order of arithmetic operations in expressions with parentheses

Expressions often contain parentheses to indicate the order in which actions should be performed. In this case a rule that specifies the order of execution of actions in expressions with parentheses, 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, the expressions in brackets are considered as components of the original expression, and they retain the order of actions already known to us. Let's look at the solutions to the examples for greater clarity.

Example.

Follow these steps 5+(7−2·3)·(6−4):2.

Solution.

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

Let's write down 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 parentheses within parentheses. There is no need to be afraid of this; you just need to consistently apply the stated rule for performing actions in expressions with brackets. Let's show the solution of the example.

Example.

Perform the operations 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 perform the actions in them first. 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 all that remains is to complete the actions: 4+24=28.

Answer:

4+(3+1+4·(2+3))=28.

In general, when an expression contains parentheses within parentheses, it is often convenient to perform actions starting with the inner parentheses and moving to the outer ones.

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

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.

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 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 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. 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 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 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.

When calculating examples, you need to follow a certain procedure. Using the rules below, we will figure out the order in which the actions are performed and what the parentheses are for.

If there are no parentheses in the expression, then:

first we perform all the operations of multiplication and division from left to right;

and then from left to right all the addition and subtraction operations.

Let's consider procedure in the following example.

We remind you that order of operations in mathematics arranged from left to right (from the beginning to the end of the example).

When calculating the value of an expression, you can record it in two ways.

First way

Each action is recorded separately with its own number under the example.
After the last action is completed, the response is necessarily written to the original example.

When calculating the results of actions with two-digit and/or three-digit numbers, be sure to list your calculations in a column.

Second way

The second method is called chain recording. All calculations are carried out in exactly the same order, but the results are written immediately after the equal sign.

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

Inside the parentheses themselves, the order of actions is the same as in expressions without parentheses.

If there are more brackets inside the brackets, then the actions inside the nested (inner) brackets are performed first.

Procedure and exponentiation

If the example contains a numeric or literal expression in brackets that must be raised to a power, then:

First we perform all the actions inside the brackets
Then we raise to a power all parentheses and numbers that stand in a power, from left to right (from the beginning to the end of the example).
We carry out the remaining steps as usual

Procedure for performing actions, rules, examples.

Numeric, alphabetic expressions and expressions with variables in their notation may contain signs of various arithmetic operations. When transforming expressions and calculating the values of expressions, actions are performed in a certain order, in other words, you must observe order of actions.

Page navigation.

First multiplication and division, then addition and subtraction

The school gives 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,
Moreover, multiplication and division are performed first, and then addition and subtraction.

Follow steps 7−3+6.

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

Indicate the order of actions in the expression 6:2·8:3.

First we divide 6 by 2, multiply this quotient by 8, and finally divide the result by 3.

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

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

The same order of operations - first multiplication and division, then addition and subtraction - should be followed when working with letter expressions.

Actions of the first and second stages

In some mathematics textbooks there is a division of arithmetic operations into operations of the first and second stages. Let's figure this out.

Actions of the first stage addition and subtraction are called, and multiplication and division are called second stage actions.

Order of arithmetic operations in expressions with parentheses

Expressions often contain parentheses to indicate the order in which actions are performed. In this case a rule that specifies the order of execution of actions in expressions with parentheses, 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.

Follow these steps 5+(7−2·3)·(6−4):2.

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

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

In general, when an expression contains parentheses within parentheses, it is often convenient to perform actions starting with the inner parentheses and moving to the outer ones.

The order of operations in expressions with roots, powers, logarithms and other functions

If the expression includes powers, roots, logarithms, sine, cosine, tangent and cotangent, as well as other functions, then their values are calculated before performing other actions, and the rules from the previous paragraphs that specify the order of actions are also taken into account. In other words, the listed things, roughly speaking, can be considered enclosed in brackets, and we know that the actions in brackets are performed first.

Let's look at the solutions to the examples.

Perform the actions in the expression (3+1)·2+6 2:3−7.

This expression contains the power of 6 2, its value must be calculated before performing other actions. So, we perform the exponentiation: 6 2 =36. We substitute this value into the original expression, it will take the form (3+1)·2+36:3−7.

Then everything is clear: we perform the actions in brackets, after which we are left with an expression without brackets, in which, in order from left to right, we first perform multiplication and division, and then addition and subtraction. We have (3+1)·2+36:3−7=4·2+36:3−7= 8+12−7=13.

You can see other, including more complex examples of performing actions in expressions with roots, powers, etc., in the article Calculating the Values of Expressions.

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Online games, simulators, presentations, lessons, encyclopedias, articles

Examples with brackets, lesson with simulators.

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.

If there are no parentheses in the example, we perform all actions in order, from left to right.
If the example contains parentheses, then first we perform the actions in brackets, and only then all other actions, starting from left to right.

*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:

If there are no parentheses in the example, first perform the operations of multiplication and division in order, from left to right. Then - the operations of addition and subtraction in order, from left to right.
If the example contains parentheses, then first we perform the operations in parentheses, then multiplication and division, and then addition and subtraction starting from left to right.

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!

1. Examples with parentheses within numbers up to 100, addition, subtraction, multiplication and division. Online trainer.

2. Mathematics simulator for grades 2 - 3 “Arrange the order of actions (letter expressions).”

3. Order of actions (we arrange the order and solve examples)

Procedure in mathematics 4th grade

Primary school is coming to an end, and soon the child will step into the advanced world of mathematics. But already during this period the student is faced with the difficulties of science. When performing a simple task, the child gets confused and lost, which ultimately leads to a negative mark for the work done. To avoid such troubles, when solving examples, you need to be able to navigate in the order in which you need to solve the example. Having distributed the actions incorrectly, the child does not complete the task correctly. The article reveals the basic rules for solving examples that contain the entire range of mathematical calculations, including brackets. Procedure in mathematics 4th grade rules and examples.

Before completing the task, ask your child to number the actions that he is going to perform. If you have any difficulties, please help.

Some rules to follow when solving examples without brackets:

If a task requires a series of operations, you must first perform division or multiplication, then addition. All actions are performed as the letter progresses. Otherwise, the result of the decision will not be correct.

If in the example you need to perform addition and subtraction, we do it in order, from left to right.

27-5+15=37 (When solving the example, we are guided by the rule. First we perform subtraction, then addition).

Teach your child to always plan and number the actions performed.

The answers to each solved action are written above the example. This will make it much easier for the child to navigate the actions.

Let's consider another option where it is necessary to distribute actions in order:

As you can see, when solving, the rule is followed: first we look for the product, then we look for the difference.

These are simple examples that require careful consideration when solving them. Many children are stunned when they see a task that contains not only multiplication and division, but also parentheses. A student who does not know the procedure for performing actions has questions that prevent him from completing the task.

As stated in the rule, first we find the product or quotient, and then everything else. But there are parentheses! What to do in this case?

Solving examples with brackets

Let's look at a specific example:

When performing this task, we first find the value of the expression enclosed in parentheses.
You should start with multiplication, then addition.
After the expression in brackets is solved, we proceed to actions outside them.
According to the rules of procedure, the next step is multiplication.
The final step will be subtraction.

As we can see in the visual example, all actions are numbered. To reinforce the topic, invite your child to solve several examples on their own:

The order in which the value of the expression should be calculated has already been arranged. The child will only have to carry out the decision directly.

Let's complicate the task. Let the child find the meaning of the expressions on his own.

7*3-5*4+(20-19) 14+2*3-(13-9)
17+2*5+(28-2) 5*3+15-(2-1*2)
24-3*2-(56-4*3) 14+12-3*(21-7)

Teach your child to solve all tasks in draft form. In this case, the student will have the opportunity to correct an incorrect decision or blots. Corrections are not allowed in the workbook. By completing tasks on their own, children see their mistakes.

Parents, in turn, should pay attention to mistakes, help the child understand and correct them. You shouldn’t overload a student’s brain with large amounts of tasks. With such actions you will discourage the child’s desire for knowledge. There should be a sense of proportion in everything.

Take a break. The child should be distracted and take a break from classes. The main thing to remember is that not everyone has a mathematical mind. Maybe your child will grow up to be a famous philosopher.

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Math lesson 2nd grade Order of actions in expressions with brackets.

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Target: 1.

3. Consolidate knowledge of the multiplication table and division by 2 – 6, the concept of divisor and

4. Learn to work in pairs in order to develop communication skills.

Equipment * : + — (), geometric material.

One, two - head up.

Three, four - arms wider.

Five, six - everyone sit down.

Seven, eight - let's discard laziness.

But first you have to find out its name. To do this you need to complete several tasks:

6 + 6 + 6 … 6 * 4 6 * 4 + 6… 6 * 5 – 6 14 dm 5 cm… 4 dm 5 cm

While we remembered the order of actions in expressions, miracles happened to the castle. We were just at the gate, and now we were in the corridor. Look, the door. And there is a castle on it. Shall we open it?

1. Subtract the quotient of 8 and 2 from the number 20.

2. Divide the difference between 20 and 8 by 2.

— How are the results different?

- Who can name the topic of our lesson?

(on massage mats)

Along the path, along the path

We gallop on our right leg,

We jump on our left leg.

Let's run along the path,

Our guess was completely correct7

Where are the actions performed first if there are parentheses in an expression?

Look at the “living examples” before us. Let's bring them to life.

* : + — ().

m – c * (a + d) + x

k: b + (a – c) * t

6. Work in pairs.

To solve them you will need geometric material.

Students complete tasks in pairs. After completion, check the work of the pairs at the board.

What new have you learned?

8. Homework.

Topic: Order of actions in expressions with brackets.

Target: 1. Derive a rule for the order of actions in expressions with brackets containing all

4 arithmetic operations,

2. To develop the ability to practically apply the rules,

4. Learn to work in pairs in order to develop communication skills.

Equipment: textbook, notebooks, cards with action signs * : + — (), geometric material.

1 .Physical exercise.

Nine, ten - sit down quietly.

2. Updating basic knowledge.

Today we are setting off on another journey through the Land of Knowledge, the city of mathematics. We have to visit one palace. Somehow I forgot its name. But let’s not be upset, you yourself can tell me its name. While I was worried, we approached the gates of the palace. Shall we come in?

1. Compare expressions:

2. Unscramble the word.

3. Statement of the problem. Discovery of something new.

So what is the name of the palace?

And when in mathematics do we talk about order?

What do you already know about the order of actions in expressions?

— Interesting, we are asked to write down and solve expressions (the teacher reads the expressions, the students write them down and solve them).

20 – 8: 2

(20 – 8) : 2

Well done. What's interesting about these expressions?

Look at the expressions and their results.

— What is common in writing expressions?

— Why do you think the results were different, since the numbers were the same?

Who would dare to formulate a rule for performing actions in expressions with brackets?

We can check the correctness of this answer in another room. Let's go there.

4. Physical exercise.

And along the same path

We will reach the mountain.

Stop. Let's rest a little

And we'll go on foot again.

5. Primary consolidation of what has been learned.

Here we are.

We need to solve two more expressions to check the correctness of our assumption.

6 * (33 – 25) 54: (6 + 3) 25 – 5 * (9 – 5) : 2

To check the correctness of the assumption, let's open the textbooks on page 33 and read the rule.

How should you perform the actions after the solution in brackets?

Letter expressions are written on the board and there are cards with action signs. * : + — (). Children go to the board one at a time, take a card with the action that needs to be done first, then the second student comes out and takes a card with the second action, etc.

a + (a – b)

a * (b + c) : d – t

m – c * ( a + d ) + x

k : b + ( a – c ) * t

(a–b) : t+d

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In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the “Achilles and the Tortoise” aporia. Here's what it sounds like:

Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.

From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.

If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia of Zeno tells about a flying arrow:

A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.

Wednesday, July 4, 2018

The differences between set and multiset are described very well on Wikipedia. Let's see.

As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.

Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.

No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.

We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.

First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...

And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.

Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.

To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."

Sunday, March 18, 2018

The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.

Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.

Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.

1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.

2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.

3. Convert individual graphic symbols into numbers. This is not a mathematical operation.

4. Add the resulting numbers. Now this is mathematics.

The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.

From a mathematical point of view, it does not matter in which number system we write a number. So, in different number systems the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With the large number 12345, I don’t want to fool my head, let’s consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.

As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.

Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.

The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.

What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.

Sign on the door He opens the door and says:

Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?

Female... The halo on top and the arrow down are male.

If such a work of design art flashes before your eyes several times a day,

Then it’s not surprising that you suddenly find a strange icon in your car:

Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.

1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.

The order of solving mathematical operations. Rules for solving examples on actions with brackets

First multiplication and division, then addition and subtraction

Actions of the first and second stages

Order of arithmetic operations in expressions with parentheses

First way

Second way

Procedure and exponentiation

Procedure for performing actions, rules, examples.

First multiplication and division, then addition and subtraction

Actions of the first and second stages

Order of arithmetic operations in expressions with parentheses

The order of operations in expressions with roots, powers, logarithms and other functions

Online games, simulators, presentations, lessons, encyclopedias, articles

Post navigation

Examples with brackets, lesson with 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

1. Examples with parentheses within numbers up to 100, addition, subtraction, multiplication and division. Online trainer.

2. Mathematics simulator for grades 2 - 3 “Arrange the order of actions (letter expressions).”

Procedure in mathematics 4th grade

Some rules to follow when solving examples without brackets:

Solving examples with brackets

Math lesson 2nd grade Order of actions in expressions with brackets.

Wednesday, July 4, 2018

Sunday, March 18, 2018