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 been able to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches; 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, it looks like time slowing down to a complete stop 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 each 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 rests 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 space at one point in time, but it is impossible to determine the fact of movement from them (naturally, additional data for calculations are still needed, trigonometry will help you). What do I want to focus on Special attention, 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 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. Applicable mathematical theory sets to the 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 entire 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 the same 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 begin to convulsively recall physics: on different coins there is different amount dirt, crystal structure and atomic arrangement of each coin is unique...
And now I have the most interest Ask: 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 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 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's not all.
From the point of view of mathematics, it does not matter in which number system we write the number. So, in different systems reckoning, 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. FROM a large number 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 were to determine the area of a rectangle in meters and centimeters, you would get exactly different results.
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 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.
The order of actions - Mathematics Grade 3 (Moro)
Short description:
In life you are constantly doing various activities: get up, wash your face, do exercises, have breakfast, go to school. Do you think this procedure can be changed? For example, have breakfast, and then wash. Probably you can. It may not be very convenient to have breakfast unwashed, but nothing terrible will happen because of this. And in mathematics, is it possible to change the order of actions at will? No, mathematics is an exact science, so even the slightest change in the order of operations will cause the answer of a numerical expression to become incorrect. In the second grade, you already got acquainted with some rules of the order of actions. So, you probably remember that parentheses govern the order in performing actions. They indicate that actions must be performed first. What other rules of procedure are there? Is the order of operations in expressions with brackets and without brackets different? You will find answers to these questions in the 3rd grade mathematics textbook when studying the topic "Order of actions". You must definitely practice applying the learned rules, and if necessary, find and correct errors in establishing the order of actions in numerical expressions. Please remember that order is important in any business, but in mathematics it has a special meaning! 
When calculating examples, you need to follow a certain procedure. With the help of the rules below, we will figure out in what order the actions are performed and what the brackets are for.
If there are no brackets in the expression, then:
Consider procedure in the next 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 evaluating the value of an expression, you can record in two ways.
First way
- Each action is recorded separately with its number under the example.
- After the last action is completed, the answer is necessarily written to the original example.
- The second method is called chaining. All calculations are carried out in exactly the same order of operations, but the results are written immediately after the equal sign.
- First, we perform all the actions inside the brackets
- Then we raise to a power all the brackets and numbers in the power, from left to right (from the beginning to the end of the example).
- Carry out the rest of the steps in the usual way
- actions are performed in order from left to right,
- where multiplication and division are performed first, and then addition and subtraction.
- If there are no brackets in the example, we perform all actions in order, from left to right.
- If the example contains parentheses, then we first perform the actions in brackets, and only then all the other actions, starting from left to right.
- If there are no brackets 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 brackets, then multiplication and division, and then addition and subtraction starting from left to right.
- 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 is subtraction.
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When calculating the results of actions with two-digit and / or three-digit numbers be sure to bring your calculations in a column.
Second way
If the expression contains parentheses, then the actions in the parentheses are performed first.
Within the parentheses themselves, the order of operations is the same as in expressions without parentheses.
If there are other 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:
The order of actions, rules, examples.
Numeric, literal and expressions with variables in their record may contain characters of various arithmetic operations. When converting expressions and 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:
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.
Follow steps 7−3+6 .
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 .
Indicate the order in which actions are performed in the expression 6:2·8:3 .
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.
First, 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 .
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 .
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.
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.
Perform the given steps 5+(7−2 3) (6−4):2 .
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 .
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.
Perform the actions in the expression 4+(3+1+4·(2+3)) .
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 .
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 .
The order in which operations are performed 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, while also taking into account the rules from the previous paragraphs that specify the order in which actions are performed. 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 consider examples.
Perform the operations in the expression (3+1) 2+6 2:3−7 .
This expression contains a power of 6 2 , its value must be calculated before performing the rest of the steps. So, we perform exponentiation: 6 2 \u003d 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 actions in brackets, after which an expression without brackets remains, 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 .
Others, including more complex examples of performing actions in expressions with roots, degrees, etc., you can see in the article calculating the values of expressions.
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Examples with brackets, a lesson with simulators.
We will look at three examples in this article:
1. Examples with brackets (addition and subtraction operations)
2. Examples with brackets (addition, subtraction, multiplication, division)
3. Examples with a lot of actions
1 Examples with brackets (addition and subtraction operations)
Let's look at three examples. In each of them, the procedure is indicated by red numbers:
We see that the order of actions in each example will be different, although the numbers and signs are the same. This is because the second and third examples have parentheses.
*This rule is for examples without multiplication and division. Rules for examples with brackets, including the operations of multiplication and division, we will consider in the second part of this article.
In order not to get confused in the example with brackets, you can turn it into a regular example, without brackets. To do this, we write the result obtained in brackets above the brackets, then we rewrite the entire example, writing this result instead of brackets, and then we perform all the actions in order, from left to right:
In simple examples, all these operations can be performed in the mind. The main thing is to first perform the action in brackets and remember the result, and then count in order, from left to right.
And now - trainers!
1) Examples with brackets up to 20. Online simulator.
2) Examples with brackets up to 100. Online simulator.
3) Examples with brackets. Trainer #2
4) Insert the missing number - examples with brackets. Training apparatus
2 Examples with brackets (addition, subtraction, multiplication, division)
Now consider examples in which, in addition to addition and subtraction, there is multiplication and division.
Let's look at examples without parentheses first:
There is one trick, how not to get confused when solving examples for the order of actions. If there are no brackets, then we perform the operations of multiplication and division, then we rewrite the example, writing down the results obtained instead of these actions. Then we perform addition and subtraction in order:
If the example contains brackets, then first you need to get rid of the brackets: rewrite the example, writing the result obtained in them instead of brackets. 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 the free signs "+" and "-" (free means not in brackets, shown by arrows in the figure).
These signs will divide our example into blocks:
Performing the actions in each block, do not forget about the procedure given above in the article. After solving each block, we perform addition and subtraction operations in order.
And now we fix the solution of the examples on the order of actions on the simulators!
1. Examples with brackets within numbers up to 100, addition, subtraction, multiplication and division. Online simulator.
2. Mathematics simulator 2 - 3 class "Arrange the order of actions (literal expressions)."
3. Order of actions (arranging the order and solving examples)
Procedure in mathematics Grade 4
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 containing the entire spectrum 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 addition. All actions are performed in the course of writing. Otherwise, the result of the solution will not be correct.
If the example requires addition and subtraction, we perform 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 - the difference.
it simple examples which require careful consideration. 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:
As we see on good 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 draft version. In this case, the student will have the opportunity to correct the wrong decision or blots. AT workbook corrections are not allowed. 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.
detskoerazvitie.info
Lesson in mathematics Grade 2 The order of actions in expressions with brackets.
Take advantage of up to 50% discounts on Infourok courses
Target: 1.
2.
3. Consolidate knowledge of the multiplication table and division by 2 - 6, the concept of a 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 need to know 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 were remembering the order of actions in expressions, miracles happened to the castle. We were just at the gate, and now we are in the corridor. Look, the door. And it has a castle. Will we open?
1. From the number 20 subtract the quotient of the numbers 8 and 2.
2. Divide the difference between the numbers 20 and 8 by 2.
- How are the results different?
Who can name the topic of our lesson?
(on massage mats)
On the track, on the track
We jump on the right leg,
We jump on the left leg.
Let's run along the path
Our assumption was completely correct7
Where are the actions performed first if there are parentheses in the expression?
See before us "live examples". Let's bring them to life.
* : + — ().
m – c * (a + d) + x
k: b + (a - c) * t
6. Work in pairs.
To solve them, you need a geometric material.
Students complete tasks in pairs. After completion, check the work of pairs at the blackboard.
What new did you learn?
8. Homework.
Topic: Order of actions in expressions with brackets.
Target: 1. Derive a rule for the order of operations in expressions with brackets containing all
4 arithmetic operations,
2. Build the ability to practical application regulations,
4. Learn to work in pairs in order to develop communication skills.
Equipment: textbook, notebooks, cards with action signs * : + — (), geometric material.
1 .Fizminutka.
Nine, ten - sit quietly.
2. Actualization of basic knowledge.
Today we are going on another journey through the country of Knowledge to 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. Let's go in?
1. Compare expressions:
2. Decipher the word.
3. Statement of the problem. Opening new.
So what is the name of the palace?
When do we talk about order in mathematics?
What do you already know about the order in which actions are performed in expressions?
- Interestingly, we are offered 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 is interesting about these expressions?
Look at expressions and their results.
- What do expressions have in common?
- Why do you think there were different results, because the numbers were the same?
Who dares 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 Minute.
And along the same path
We will reach the mountain.
Stop. Let's get some rest
And let's go on foot again.
5. Primary consolidation of the studied.
Here we come.
We need to solve two more expressions to check if our guess is correct.
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 actions after the solution in parentheses?
Alphabetic expressions are written on the board and cards with action signs are lying. * : + — (). 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 – c)
a * (b + c) : d – t
m – c * ( a + d ) + x
k : b + ( a – c ) * t
(a-b) : t + d
6. Work in pairs.
Knowing the order of actions is necessary not only for solving examples, but also when solving problems, we also encounter this rule. Now you will see this by working in pairs. You will need to solve problems from #3 page 33.
7. Bottom line.
Which palace did you and I travel to today?
Did you like the lesson?
How to perform operations in expressions with brackets?
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. Math Lessons: Guidelines for the teacher. 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. Maths: Verification work. Grade 3 - M.: Education, 2012.
- V.N. Rudnitskaya. Tests. - M.: "Exam", 2012.
- Festival.1september.ru ().
- Sosnovoborsk-soobchestva.ru ().
- Openclass.ru ().
Homework
1. Determine the order of actions in these expressions. Find the meaning of expressions.
2. Determine in 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.
The video tutorial "Procedure for performing actions" explains in detail important topic mathematics - the sequence of arithmetic operations when solving an expression. During the video lesson, it is considered what priority various mathematical operations have, how it is used in the calculation of expressions, examples are given for mastering the material, the knowledge gained is summarized in solving tasks, where all the considered operations are present. With the help of a video lesson, the teacher has the opportunity to quickly achieve the goals of the lesson, increase its effectiveness. The video can be used as a visual material accompanying the teacher's explanation, as well as an independent part of the lesson.
The visual material uses techniques that help to better achieve understanding of the topic, as well as to remember important rules. With the help of color and different spelling, the features and properties of operations are highlighted, the features of solving examples are noted. Animation effects help serve consistently educational material and draw students' attention to important points. The video is voiced, therefore it is supplemented with teacher's comments that help the student understand and remember the topic.
The video tutorial starts by introducing the topic. Then it is noted that multiplication, subtraction are operations of the first stage, the operations of multiplication and division are called operations of the second stage. This definition will need to be operated further, displayed on the screen and highlighted in large colored print. Then the rules that make up the order in which operations are performed are presented. The first order rule is displayed, which indicates that if there are no brackets in the expression, if there are actions of one stage, these actions must be performed in order. The second rule of order states that if there are actions of both stages and there are no brackets, the operations of the second stage are performed first, then the operations of the first stage are performed. The third rule establishes the order in which operations are performed for expressions that include parentheses. It is noted that in this case operations in parentheses are performed first. The wording of the rules is highlighted in color and recommended for memorization.
Next, it is proposed to learn the order of operations, considering examples. The solution of an expression containing only operations of addition and subtraction is described. The main features that affect the order of calculations are noted - there are no brackets, there are operations of the first stage. Below is a step-by-step description of how calculations are performed, first subtraction, then addition twice, and then subtraction.
In the second example 780:39·212:156·13 it is required to evaluate the expression by performing actions according to the order. It is noted that this expression contains only operations of the second stage, without brackets. AT this example All actions are performed strictly from left to right. Below, the actions are painted in turn, gradually approaching the answer. The result of the calculation is the number 520.
In the third example, the solution of the example is considered, in which there are operations of both stages. It is noted that in this expression there are no brackets, but there are actions of both steps. According to the order of operations, operations of the second stage are performed, after that - operations of the first stage. Below, the solution is described by actions, in which three operations are performed first - multiplication, division, one more division. Then, with the found values of the product and quotients, operations of the first stage are performed. During the solution, curly brackets combine the actions of each step for clarity.
The following example contains parentheses. Therefore, it is shown that the first calculations are performed on the expressions in brackets. After them, operations of the second stage are performed, followed by the first.
The following is a note on when you can not write parentheses when solving expressions. It is noted that this is possible only in the case when the elimination of parentheses does not change the order of operations. An example is the expression with brackets (53-12)+14, which contains only the operations of the first stage. By rewriting 53-12+14 with the parentheses removed, it can be noted that the search order for the value will not change - first subtract 53-12=41, and then add 41+14=55. It is noted below that you can change the order of operations when finding a solution to an expression using the properties of operations.
At the end of the video lesson, the studied material is summarized in the conclusion that each expression that needs to be solved defines a specific program for calculation, consisting of commands. An example of such a program is presented in the description of the solution complex example, which is the quotient of (814+36 27) and (101-2052:38). The specified program contains the following steps: 1) find the product of 36 with 27, 2) add the found sum to 814, 3) divide the number 2052 by 38, 4) subtract the result of dividing 3 points from the number 101, 5) divide the result of step 2 by the result of step four.
At the end of the video lesson there is a list of questions that students are asked to answer. Among them are the ability to distinguish between the actions of the first and second stages, questions about the order in which actions are performed in expressions with actions of the same stage and different stages, and the order in which actions are performed when there are brackets in the expression.
The video lesson "Procedure for performing actions" is recommended to be used in a traditional school lesson to increase the effectiveness of the lesson. Also visual material will be useful for distance learning. If the student needs an additional lesson to master the topic or he studies it on his own, the video can be recommended for self-study.
