Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about 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 universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.
From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, this looks like a slowdown in time until it stops completely at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.
If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its 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 infinitely quickly overtake the tortoise."
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise crawls 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 one hundred steps. Now Achilles is eight hundred paces 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 insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet 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 of 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 the flying arrow is at rest at different points in space, which, in fact, is movement. There is another point 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 the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.
Wednesday, July 4, 2018
Very well the differences between set and multiset are described in Wikipedia. We look.
As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.
Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of 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 desk, paying salaries. Here a mathematician comes to us for his money. We count the whole amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.
First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will frantically recall physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms for each coin is unique ...
And now I have the most interesting question: where is the boundary beyond which 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 here is not even close.
Look here. We select football stadiums with the same field area. The area of the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us about either 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 will 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 they are shamans for that, 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 "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can 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 elementarily.
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's say we 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 to a number graphic symbol. This is not a mathematical operation.
2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic characters to numbers. This is not a mathematical operation.
4. Add up the resulting numbers. Now that's mathematics.
The sum of the digits of the number 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that is not all.
From the point of view of mathematics, it does not matter in which number system we write the 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 a large number of 12345, I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have 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 would get completely different results when determining the area of a rectangle in meters and centimeters.
Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement of 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 action does not depend on the value of the number, the unit of measurement used, and on who performs this action.
Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?
Female... A halo on top and an arrow down is male.
If you have such a work of design art flashing before your eyes several times a day,
Then it is not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.
1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.
Lesson topic: "The order of execution of actions in expressions without brackets and with brackets.
The purpose of the lesson: create conditions for consolidating the ability to apply knowledge about the order of performing actions in expressions without brackets and with brackets in various situations, the ability to solve problems with an expression.
Lesson objectives.
Educational:
To consolidate students' knowledge about the rules for performing actions in expressions without brackets and with brackets; to form their ability to use these rules when calculating specific expressions; improve computing skills; repeat the tabular cases of multiplication and division;
Developing:
Develop computational skills, logical thinking, attention, memory, cognitive abilities of students,
communication skills;
Educational:
Cultivate a tolerant attitude towards each other, mutual cooperation,
culture of behavior in the classroom, accuracy, independence, to cultivate interest in mathematics.
Formed UUD:
Regulatory UUD:
work according to the proposed plan, instructions;
put forward their hypotheses on the basis of educational material;
exercise self-control.
Cognitive UUD:
know the order of operations:
be able to explain their content;
understand the rule of order of actions;
find the values of expressions according to the rules of the order of execution;
actions, using text tasks for this;
write the solution of the problem with an expression;
apply rules for the order of actions;
be able to apply the acquired knowledge in the performance of control work.
Communicative UUD:
listen and understand the speech of others;
express their thoughts with sufficient completeness and accuracy;
allow the possibility of different points of view, strive to understand the position of the interlocutor;
work in a team of different content (pair, small group, whole class), participate in discussions, working in pairs;
Personal UUD:
establish a relationship between the purpose of the activity and its result;
define rules of conduct common to all;
to express the ability for self-assessment on the basis of the criterion of success of educational activity.
Planned result:
Subject:
Know the rules for ordering actions.
Be able to explain their content.
Be able to solve problems using expressions.
Personal:
Be able to conduct self-assessment based on the criterion of success of educational activities.
Metasubject:
Be able to determine and formulate the goal in the lesson with the help of a teacher; pronounce the sequence of actions in the lesson; work according to a collective plan; evaluate the correctness of the action at the level of an adequate retrospective assessment; plan your action in accordance with the task; make the necessary adjustments to the action after its completion, based on its assessment and taking into account the nature of the errors made; make one's guess Regulatory UUD ).
Be able to formulate your thoughts orally; listen and understand the speech of others; jointly agree on the rules of behavior and communication at school and follow them ( Communicative UUD ).
To be able to navigate in their system of knowledge: to distinguish the new from the already known with the help of a teacher; acquire new knowledge: find answers to questions using a textbook, your life experience and information received in the lesson (Cognitive UUD ).
During the classes
1. Organizational moment.
To make our lesson brighter,
We will share the good.
Stretch out your palms
Put your love in them
And smile at each other.
Take your jobs.
They opened notebooks, wrote down the date and classwork.
2. Actualization of knowledge.
In the lesson, we will have to consider in detail the order in which arithmetic operations are performed in expressions without brackets and with brackets.
Verbal counting.
Find the right answer game.
(Each student has a sheet with numbers)
I read the assignments, and you, having completed the actions in your mind, must cross out the result, that is, the answer, with a cross.
I conceived a number, subtracted 80 from it, got 18. What number did I conceive? (98)
I conceived a number, added 12 to it, got 70. What number did I conceive? (58)
The first term is 90, the second term is 12. Find the sum. (102)
Connect your results.
What geometry did you get? (Triangle)
Tell us what you know about this geometric figure. (Has 3 sides, 3 tops, 3 corners)
We continue to work on the card.
Find the difference between the numbers 100 and 22 . (78)
Reduced 99, subtracted 19. Find the difference. (80).
Take the number 25 4 times. (100)
Draw 1 more triangle inside the triangle, connecting the results.
How many triangles did you get? (5)
3. Work on the topic of the lesson. Observing the change in the value of an expression depending on the order in which arithmetic operations are performed
In life, we constantly perform some kind of action: we walk, study, read, write, count, smile, quarrel and make up. We perform these steps in a different order. Sometimes they can be swapped, sometimes they can't. For example, going to school in the morning, you can first do exercises, then make the bed, or vice versa. But you can’t go to school first and then put on clothes.
And in mathematics, is it necessary to perform arithmetic operations in a certain order?
Let's check
Let's compare the expressions:
8-3+4 and 8-3+4
We see that both expressions are exactly the same.
Let's execute actions in one expression from left to right, and in another from right to left. Numbers can indicate the order in which actions are performed (Fig. 1).
Rice. 1. Procedure
In the first expression, we will first perform the subtraction operation, and then add the number 4 to the result.
In the second expression, we first find the value of the sum, and then subtract the result 7 from 8.
We see that the values of the expressions are different.
Let's conclude: The order in which arithmetic operations are performed cannot be changed..
Arithmetic order in expressions without brackets
Let's learn the rule for performing arithmetic operations in expressions without brackets.
If the expression without brackets includes only addition and subtraction, or only multiplication and division, then the actions are performed in the order in which they are written.
Let's practice.
Consider the expression
This expression has only addition and subtraction operations. These actions are called first step actions.
We perform actions from left to right in order (Fig. 2).
Rice. 2. Procedure
Consider the second expression
In this expression, there are only operations of multiplication and division - These are the second step actions.
We perform actions from left to right in order (Fig. 3).
Rice. 3. Procedure
In what order are arithmetic operations performed if the expression contains not only addition and subtraction, but also multiplication and division?
If the expression without brackets includes not only addition and subtraction, but also multiplication and division, or both of these operations, then first perform multiplication and division in order (from left to right), and then addition and subtraction.
Consider an expression.
We reason like this. This expression contains the operations of addition and subtraction, multiplication and division. We act according to the rule. First, we perform in order (from left to right) multiplication and division, and then addition and subtraction. Let's lay out the procedure.
Let's calculate the value of the expression.
18:2-2*3+12:3=9-6+4=3+4=7
Order of execution of arithmetic operations in expressions with brackets
In what order are arithmetic operations performed if the expression contains parentheses?
If the expression contains parentheses, then the value of the expressions in the parentheses is calculated first.
Consider an expression.
30 + 6 * (13 - 9)
We see that in this expression there is an action in brackets, which means that we will perform this action first, then, in order, multiplication and addition. Let's lay out the procedure.
30 + 6 * (13 - 9)
Let's calculate the value of the expression.
30+6*(13-9)=30+6*4=30+24=54
The rule for performing arithmetic operations in expressions without brackets and with brackets
How should one reason in order to correctly establish the order of arithmetic operations in a numerical expression?
Before proceeding with the calculations, it is necessary to consider the expression (find out if it contains brackets, what actions it has) and only after that perform the actions in the following order:
1. actions written in brackets;
2. multiplication and division;
3. addition and subtraction.
The diagram will help you remember this simple rule (Fig. 4).
Rice. 4. Procedure
4. Consolidation Fulfillment of training tasks for the learned rule
Let's practice.
Consider the expressions, establish the order of operations and perform the calculations.
43 - (20 - 7) +15
32 + 9 * (19 - 16)
Let's follow the rules. The expression 43 - (20 - 7) +15 has operations in parentheses, as well as operations of addition and subtraction. Let's set the course of action. The first step is to perform the action in brackets, and then in order from left to right, subtraction and addition.
43 - (20 - 7) +15 =43 - 13 +15 = 30 + 15 = 45
The expression 32 + 9 * (19 - 16) has operations in parentheses, as well as operations of multiplication and addition. According to the rule, we first perform the action in brackets, then multiplication (the number 9 is multiplied by the result obtained by subtraction) and addition.
32 + 9 * (19 - 16) =32 + 9 * 3 = 32 + 27 = 59
In the expression 2*9-18:3 there are no brackets, but there are operations of multiplication, division and subtraction. We act according to the rule. First, we perform multiplication and division from left to right, and then from the result obtained by multiplication, we subtract the result obtained by division. That is, the first action is multiplication, the second is division, and the third is subtraction.
2*9-18:3=18-6=12
Let's find out if the order of actions in the following expressions is defined correctly.
37 + 9 - 6: 2 * 3 =
18: (11 - 5) + 47=
7 * 3 - (16 + 4)=
We reason like this.
37 + 9 - 6: 2 * 3 =
There are no brackets in this expression, which means that we first perform multiplication or division from left to right, then addition or subtraction. In this expression, the first action is division, the second is multiplication. The third action should be addition, the fourth - subtraction. Conclusion: the order of actions is defined correctly.
Find the value of this expression.
37+9-6:2*3 =37+9-3*3=37+9-9=46-9=37
We continue to argue.
The second expression contains brackets, which means that we first perform the action in brackets, then from left to right multiplication or division, addition or subtraction. We check: the first action is in brackets, the second is division, the third is addition. Conclusion: the order of actions is defined incorrectly. Correct the errors, find the value of the expression.
18:(11-5)+47=18:6+47=3+47=50
This expression also contains brackets, which means that we first perform the action in brackets, then from left to right multiplication or division, addition or subtraction. We check: the first action is in brackets, the second is multiplication, the third is subtraction. Conclusion: the order of actions is defined incorrectly. Correct the errors, find the value of the expression.
7*3-(16+4)=7*3-20=21-20=1
Let's complete the task.
Let's arrange the order of actions in the expression using the studied rule (Fig. 5).
Rice. 5. Procedure
We do not see numerical values, so we will not be able to find the meaning of expressions, but we will practice applying the learned rule.
We act according to the algorithm.
The first expression has parentheses, so the first action is in parentheses. Then from left to right multiplication and division, then from left to right subtraction and addition.
The second expression also contains brackets, which means that we perform the first action in brackets. After that, from left to right, multiplication and division, after that - subtraction.
Let's check ourselves (Fig. 6).
Rice. 6. Procedure
5. Summarizing.
Today in the lesson we got acquainted with the rule of the order of execution of actions in expressions without brackets and with brackets. In the course of completing tasks, we determined whether the meaning of expressions depends on the order in which arithmetic operations are performed, found out whether the order of arithmetic operations differs in expressions without brackets and with brackets, practiced applying the learned rule, searched for and corrected mistakes made in determining the order of actions.
And when calculating the values of expressions, actions are performed in a certain order, in other words, you must observe order of actions.
In this article, we will figure out which actions should be performed first, and which ones after them. Let's start with the simplest cases, when the expression contains only numbers or variables connected by plus, minus, multiply and divide. Next, we will explain what order of execution of actions should be followed in expressions with brackets. Finally, consider the sequence in which actions are performed in expressions containing powers, roots, and other functions.
Page navigation.
First multiplication and division, then addition and subtraction
The school provides the following a rule that determines the order in which actions are performed in expressions without parentheses:
- actions are performed in order from left to right,
- where multiplication and division are performed first, and then addition and subtraction.
The stated rule is perceived quite naturally. Performing actions in order from left to right is explained by the fact that it is customary for us to keep records from left to right. And the fact that multiplication and division is performed before addition and subtraction is explained by the meaning that these actions carry in themselves.
Let's look at a few examples of the application of this rule. For examples, we will take the simplest numerical expressions so as not to be distracted by calculations, but to focus on the order in which actions are performed.
Example.
Follow steps 7−3+6 .
Decision.
The original expression does not contain parentheses, nor does it contain multiplication and division. Therefore, we should perform all actions in order from left to right, that is, first we subtract 3 from 7, we get 4, after which we add 6 to the resulting difference 4, we get 10.
Briefly, the solution can be written as follows: 7−3+6=4+6=10 .
Answer:
7−3+6=10 .
Example.
Indicate the order in which actions are performed in the expression 6:2·8:3 .
Decision.
To answer the question of the problem, let's turn to the rule that indicates the order in which actions are performed in expressions without brackets. The original expression contains only the operations of multiplication and division, and according to the rule, they must be performed in order from left to right.
Answer:
At first 6 divided by 2, this quotient is multiplied by 8, finally, the result is divided by 3.
Example.
Calculate the value of the expression 17−5·6:3−2+4:2 .
Decision.
First, let's determine in what order the actions in the original expression should be performed. It includes both multiplication and division and addition and subtraction. First, from left to right, you need to perform multiplication and division. So we multiply 5 by 6, we get 30, we divide this number by 3, we get 10. Now we divide 4 by 2, we get 2. We substitute the found value 10 instead of 5 6:3 in the original expression, and the value 2 instead of 4:2, we have 17−5 6:3−2+4:2=17−10−2+2.
There is no multiplication and division in the resulting expression, so it remains to perform the remaining actions in order from left to right: 17−10−2+2=7−2+2=5+2=7 .
Answer:
17−5 6:3−2+4:2=7 .
At first, in order not to confuse the order of performing actions when calculating the value of an expression, it is convenient to place numbers above the signs of actions corresponding to the order in which they are performed. For the previous example, it would look like this: .
The same order of operations - first multiplication and division, then addition and subtraction - should be followed when working with literal expressions.
Steps 1 and 2
In some textbooks on mathematics, there is a division of arithmetic operations into operations of the first and second steps. Let's deal with this.
Definition.
First step actions are called addition and subtraction, and multiplication and division are called second step actions.
In these terms, the rule from the previous paragraph, which determines the order in which actions are performed, will be written as follows: if the expression does not contain brackets, then in order from left to right, the actions of the second stage (multiplication and division) are performed first, then the actions of the first stage (addition and subtraction).
Order of execution of arithmetic operations in expressions with brackets
Expressions often contain parentheses to indicate the order in which the actions are to be performed. In this case a rule that specifies the order in which actions are performed in expressions with brackets, is formulated as follows: first, the actions in brackets are performed, while multiplication and division are also performed in order from left to right, then addition and subtraction.
So, expressions in brackets are considered as components of the original expression, and the order of actions already known to us is preserved in them. Consider the solutions of examples for greater clarity.
Example.
Perform the given steps 5+(7−2 3) (6−4):2 .
Decision.
The expression contains brackets, so let's first perform the operations in the expressions enclosed in these brackets. Let's start with the expression 7−2 3 . In it, you must first perform the multiplication, and only then the subtraction, we have 7−2 3=7−6=1 . We pass to the second expression in brackets 6−4 . There is only one action here - subtraction, we perform it 6−4=2 .
We substitute the obtained values into the original expression: 5+(7−2 3)(6−4):2=5+1 2:2. In the resulting expression, first we perform multiplication and division from left to right, then subtraction, we get 5+1 2:2=5+2:2=5+1=6 . On this, all actions are completed, we adhered to the following order of their execution: 5+(7−2 3) (6−4):2 .
Let's write a short solution: 5+(7−2 3)(6−4):2=5+1 2:2=5+1=6.
Answer:
5+(7−2 3)(6−4):2=6 .
It happens that an expression contains brackets within brackets. You should not be afraid of this, you just need to consistently apply the voiced rule for performing actions in expressions with brackets. Let's show an example solution.
Example.
Perform the actions in the expression 4+(3+1+4·(2+3)) .
Decision.
This is an expression with brackets, which means that the execution of actions must begin with the expression in brackets, that is, with 3+1+4 (2+3) . This expression also contains parentheses, so you must first perform actions in them. Let's do this: 2+3=5 . Substituting the found value, we get 3+1+4 5 . In this expression, we first perform multiplication, then addition, we have 3+1+4 5=3+1+20=24 . The initial value, after substituting this value, takes the form 4+24 , and it remains only to complete the actions: 4+24=28 .
Answer:
4+(3+1+4 (2+3))=28 .
In general, when parentheses within parentheses are present in an expression, it is often convenient to start with the inner parentheses and work your way to the outer ones.
For example, let's say we need to perform operations in the expression (4+(4+(4−6:2))−1)−1 . First, we perform actions in internal brackets, since 4−6:2=4−3=1 , then after that the original expression will take the form (4+(4+1)−1)−1 . Again, we perform the action in the inner brackets, since 4+1=5 , then we arrive at the following expression (4+5−1)−1 . Again, we perform the actions in brackets: 4+5−1=8 , while we arrive at the difference 8−1 , which is equal to 7 .
In this lesson, the procedure for performing arithmetic operations in expressions without brackets and with brackets is considered in detail. Students are given the opportunity in the course of completing assignments to determine whether the meaning of expressions depends on the order in which arithmetic operations are performed, to find out whether the order of arithmetic operations differs in expressions without brackets and with brackets, to practice applying the learned rule, to find and correct errors made in determining the order of actions.
In life, we constantly perform some kind of action: we walk, study, read, write, count, smile, quarrel and make up. We perform these steps in a different order. Sometimes they can be swapped, sometimes they can't. For example, going to school in the morning, you can first do exercises, then make the bed, or vice versa. But you can’t go to school first and then put on clothes.
And in mathematics, is it necessary to perform arithmetic operations in a certain order?
Let's check
Let's compare the expressions:
8-3+4 and 8-3+4
We see that both expressions are exactly the same.
Let's execute actions in one expression from left to right, and in another from right to left. Numbers can indicate the order in which actions are performed (Fig. 1).
Rice. 1. Procedure
In the first expression, we will first perform the subtraction operation, and then add the number 4 to the result.
In the second expression, we first find the value of the sum, and then subtract the result 7 from 8.
We see that the values of the expressions are different.
Let's conclude: The order in which arithmetic operations are performed cannot be changed..
Let's learn the rule for performing arithmetic operations in expressions without brackets.
If the expression without brackets includes only addition and subtraction, or only multiplication and division, then the actions are performed in the order in which they are written.
Let's practice.
Consider the expression
This expression has only addition and subtraction operations. These actions are called first step actions.
We perform actions from left to right in order (Fig. 2).
Rice. 2. Procedure
Consider the second expression
In this expression, there are only operations of multiplication and division - These are the second step actions.
We perform actions from left to right in order (Fig. 3).
Rice. 3. Procedure
In what order are arithmetic operations performed if the expression contains not only addition and subtraction, but also multiplication and division?
If the expression without brackets includes not only addition and subtraction, but also multiplication and division, or both of these operations, then first perform multiplication and division in order (from left to right), and then addition and subtraction.
Consider an expression.
We reason like this. This expression contains the operations of addition and subtraction, multiplication and division. We act according to the rule. First, we perform in order (from left to right) multiplication and division, and then addition and subtraction. Let's lay out the procedure.
Let's calculate the value of the expression.
18:2-2*3+12:3=9-6+4=3+4=7
In what order are arithmetic operations performed if the expression contains parentheses?
If the expression contains parentheses, then the value of the expressions in the parentheses is calculated first.
Consider an expression.
30 + 6 * (13 - 9)
We see that in this expression there is an action in brackets, which means that we will perform this action first, then, in order, multiplication and addition. Let's lay out the procedure.
30 + 6 * (13 - 9)
Let's calculate the value of the expression.
30+6*(13-9)=30+6*4=30+24=54
How should one reason in order to correctly establish the order of arithmetic operations in a numerical expression?
Before proceeding with the calculations, it is necessary to consider the expression (find out if it contains brackets, what actions it has) and only after that perform the actions in the following order:
1. actions written in brackets;
2. multiplication and division;
3. addition and subtraction.
The diagram will help you remember this simple rule (Fig. 4).
Rice. 4. Procedure
Let's practice.
Consider the expressions, establish the order of operations and perform the calculations.
43 - (20 - 7) +15
32 + 9 * (19 - 16)
Let's follow the rules. The expression 43 - (20 - 7) +15 has operations in parentheses, as well as operations of addition and subtraction. Let's set the course of action. The first step is to perform the action in brackets, and then in order from left to right, subtraction and addition.
43 - (20 - 7) +15 =43 - 13 +15 = 30 + 15 = 45
The expression 32 + 9 * (19 - 16) has operations in parentheses, as well as operations of multiplication and addition. According to the rule, we first perform the action in brackets, then multiplication (the number 9 is multiplied by the result obtained by subtraction) and addition.
32 + 9 * (19 - 16) =32 + 9 * 3 = 32 + 27 = 59
In the expression 2*9-18:3 there are no brackets, but there are operations of multiplication, division and subtraction. We act according to the rule. First, we perform multiplication and division from left to right, and then from the result obtained by multiplication, we subtract the result obtained by division. That is, the first action is multiplication, the second is division, and the third is subtraction.
2*9-18:3=18-6=12
Let's find out if the order of actions in the following expressions is defined correctly.
37 + 9 - 6: 2 * 3 =
18: (11 - 5) + 47=
7 * 3 - (16 + 4)=
We reason like this.
37 + 9 - 6: 2 * 3 =
There are no brackets in this expression, which means that we first perform multiplication or division from left to right, then addition or subtraction. In this expression, the first action is division, the second is multiplication. The third action should be addition, the fourth - subtraction. Conclusion: the order of actions is defined correctly.
Find the value of this expression.
37+9-6:2*3 =37+9-3*3=37+9-9=46-9=37
We continue to argue.
The second expression contains brackets, which means that we first perform the action in brackets, then from left to right multiplication or division, addition or subtraction. We check: the first action is in brackets, the second is division, the third is addition. Conclusion: the order of actions is defined incorrectly. Correct the errors, find the value of the expression.
18:(11-5)+47=18:6+47=3+47=50
This expression also contains brackets, which means that we first perform the action in brackets, then from left to right multiplication or division, addition or subtraction. We check: the first action is in brackets, the second is multiplication, the third is subtraction. Conclusion: the order of actions is defined incorrectly. Correct the errors, find the value of the expression.
7*3-(16+4)=7*3-20=21-20=1
Let's complete the task.
Let's arrange the order of actions in the expression using the studied rule (Fig. 5).
Rice. 5. Procedure
We do not see numerical values, so we will not be able to find the meaning of expressions, but we will practice applying the learned rule.
We act according to the algorithm.
The first expression has parentheses, so the first action is in parentheses. Then from left to right multiplication and division, then from left to right subtraction and addition.
The second expression also contains brackets, which means that we perform the first action in brackets. After that, from left to right, multiplication and division, after that - subtraction.
Let's check ourselves (Fig. 6).
Rice. 6. Procedure
Today in the lesson we got acquainted with the rule of the order of execution of actions in expressions without brackets and with brackets.
Bibliography
- M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Enlightenment", 2012.
- M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Enlightenment", 2012.
- M.I. Moreau. Mathematics lessons: Guidelines for teachers. Grade 3 - M.: Education, 2012.
- Regulatory document. Monitoring and evaluation of learning outcomes. - M.: "Enlightenment", 2011.
- "School of Russia": Programs for elementary school. - M.: "Enlightenment", 2011.
- S.I. Volkov. Mathematics: Testing 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 which expression this order of actions is performed:
1. multiplication; 2. division;. 3. addition; 4. subtraction; 5. addition. Find the value of this expression.
3. Compose three expressions in which the following order of actions is performed:
1. multiplication; 2. addition; 3. subtraction
1. addition; 2. subtraction; 3. addition
1. multiplication; 2. division; 3. addition
Find the meaning of these expressions.
Primary school is coming to an end, soon the child will step into the in-depth world of mathematics. But already in this period, the student is faced with the difficulties of science. Performing a simple task, the child gets confused, lost, which as a result leads to a negative mark for the work performed. 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. Incorrectly distributing actions, the child does not correctly perform the task. The article reveals the basic rules for solving examples that contain the whole range of mathematical calculations, including brackets. The order of actions in mathematics grade 4 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 needs to perform a series of actions, you must first perform division or multiplication, then. All actions are performed in the course of writing. Otherwise, the result of the solution will not be correct.
If in the example it is required to execute, we execute 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 to be performed.
The answers to each solved action are written above the example. So it will be much easier for the child to navigate the actions.
Consider another option where it is necessary to distribute the actions in order:
As you can see, when solving, the rule is observed, first we look for the product, after that - the difference.
These are simple examples that require attention to solve. Many children fall into a stupor at the sight of a task in which there is not only multiplication and division, but also brackets. A student who does not know the order of performing actions has questions that prevent him from completing the task.
As stated in the rule, first we find a work or a particular, and then everything else. But then there are brackets! How to proceed in this case?
Solving examples with brackets
Let's take a specific example:
- When performing this task, first find the value of the expression enclosed in brackets.
- Start with multiplication, then add.
- After the expression in the brackets is solved, we proceed to the actions outside them.
- According to the order of operations, the next step is multiplication.
- The final step will be.
As you can see in the illustrative example, all actions are numbered. To consolidate the topic, invite the child to solve several examples on his own:
The order in which the value of the expression should be evaluated is already set. The child will only have to execute the decision directly.
Let's complicate the task. Let the child find the meaning of the expressions on their 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 a draft version. In this case, the student will have the opportunity to correct the wrong decision or blots. Corrections are not allowed in the workbook. When doing tasks on their own, children see their mistakes.
Parents, in turn, should pay attention to mistakes, help the child understand and correct them. Do not load the student's brain with large volumes of tasks. By such actions, you will beat off the child's desire for knowledge. There must be a sense of proportion in everything.
Take a break. The child should be distracted and rest from classes. The main thing to remember is that not everyone has a mathematical mindset. Maybe your child will grow up to be a famous philosopher.
