Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells about a flying arrow:
A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia logical paradox it can be overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to 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 is still needed for calculations, trigonometry will help you). What I want to point out 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 research.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...
And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.
Let's figure out what and how we do in order to find the sum of numbers given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now this is mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different number systems the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. WITH a large number 12345 I don’t want to fool my head, let’s look at the number 26 from the article about . Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.
As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, it means it has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
October 24th, 2017 admin
Lopatko Irina Georgievna
Target: formation of knowledge about the order of performing arithmetic operations in numerical expressions without brackets and with brackets, consisting of 2-3 actions.
Tasks:
Educational: to develop in students the ability to use the rules of the order of actions when calculating specific expressions, the ability to apply an algorithm of actions.
Developmental: develop skills of working in pairs, mental activity of students, the ability to reason, compare and contrast, calculation skills and mathematical speech.
Educational: cultivate interest in the subject, tolerant attitude towards each other, mutual cooperation.
Type: learning new material
Equipment: presentation, visuals, handouts, cards, textbook.
Methods: verbal, visual and figurative.
DURING THE CLASSES
- Organizing time
Greetings.
We came here to study
Don't be lazy, but work.
We work diligently
Let's listen carefully.
Markushevich said great words: “Whoever studies mathematics from childhood develops attention, trains his brain, his will, cultivates perseverance and perseverance in achieving goals.” Welcome to math lesson!
- Updating knowledge
The subject of mathematics is so serious that no opportunity should be missed to make it more entertaining.(B. Pascal)
I suggest you complete logical tasks. You are ready?
Which two numbers, when multiplied, give the same result as when added? (2 and 2)
From under the fence you can see 6 pairs of horse legs. How many of these animals are there in the yard? (3)
A rooster standing on one leg weighs 5 kg. How much will he weigh standing on two legs? (5kg)
There are 10 fingers on the hands. How many fingers are there on 6 hands? (thirty)
The parents have 6 sons. Everyone has a sister. How many children are there in the family? (7)
How many tails do seven cats have?
How many noses do two dogs have?
How many ears do 5 babies have?
Guys, this is exactly the kind of work I expected from you: you were active, attentive, and smart.
Assessment: verbal.
Verbal counting
BOX OF KNOWLEDGE
Product of numbers 2 * 3, 4 * 2;
Partial numbers 15: 3, 10:2;
Sum of numbers 100 + 20, 130 + 6, 650 + 4;
The difference between numbers is 180 – 10, 90 – 5, 340 – 30.
Components of multiplication, division, addition, subtraction.
Assessment: students independently evaluate each other
- Communicating the topic and purpose of the lesson
“To digest knowledge, you need to absorb it with appetite.”(A. Franz)
Are you ready to absorb knowledge with appetite?
Guys, Masha and Misha were offered such a chain
24 + 40: 8 – 4=
Masha decided it like this:
24 + 40: 8 – 4= 25 correct? Children's answers.
And Misha decided like this:
24 + 40: 8 – 4= 4 correct? Children's answers.
What surprised you? It seems that both Masha and Misha decided correctly. Then why do they have different answers?
They counted in different orders; they did not agree in what order they would count.
What does the calculation result depend on? From order.
What do you see in these expressions? Numbers, signs.
What are signs called in mathematics? Actions.
What order did the guys not agree on? About the procedure.
What will we study in class? What is the topic of the lesson?
We will study the order of arithmetic operations in expressions.
Why do we need to know the procedure? Perform calculations correctly in long expressions
"Basket of Knowledge". (The basket hangs on the board)
Students name associations related to the topic.
- Learning new material
Guys, please listen to what the French mathematician D. Poya said: “The best way to study something is to discover it for yourself.” Are you ready for discoveries?
180 – (9 + 2) =
Read the expressions. Compare them.
How are they similar? 2 actions, same numbers
What is the difference? Parentheses, different actions
Rule 1.
Read the rule on the slide. Children read the rule aloud.
In expressions without parentheses containing only addition and subtraction or multiplication and division, operations are performed in the order they are written: from left to right.
What actions are we talking about here? +, — or : , ·
From these expressions, find only those that correspond to rule 1. Write them down in your notebook.
Calculate the values of the expressions.
Examination.
180 – 9 + 2 = 173
Rule 2.
Read the rule on the slide.
Children read the rule aloud.
In expressions without parentheses, multiplication or division are performed first, in order from left to right, and then addition or subtraction.
:, · and +, — (together)
Are there parentheses? No.
What actions will we perform first? ·, : from left to right
What actions will we take next? +, — left, right
Find their meanings.
Examination.
180 – 9 * 2 = 162
Rule 3
In expressions with parentheses, first evaluate the value of the expressions in parentheses, thenmultiplication or division are performed in order from left to right, and then addition or subtraction.
What arithmetic operations are indicated here?
:, · and +, — (together)
Are there parentheses? Yes.
What actions will we perform first? In brackets
What actions will we take next? ·, : from left to right
And then? +, — left, right
Write down expressions that relate to the second rule.
Find their meanings.
Examination.
180: (9 * 2) = 10
180 – (9 + 2) = 169
Once again, we all say the rule together.
PHYSMINUTE
- Consolidation
“Much of mathematics does not remain in the memory, but when you understand it, then it is easy to remember what you have forgotten on occasion.”, said M.V. Ostrogradsky. Now we will remember what we just learned and apply new knowledge in practice .
Page 52 No. 2
(52 – 48) * 4 =
Page 52 No. 6 (1)
The students collected 700 kg of vegetables in the greenhouse: 340 kg of cucumbers, 150 kg of tomatoes, and the rest - peppers. How many kilograms of peppers did the students collect?
What are they talking about? What is known? What do you need to find?
Let's try to solve this problem with an expression!
700 – (340 + 150) = 210 (kg)
Answer: The students collected 210 kg of pepper.
Work in pairs.
Cards with the task are given.
5 + 5 + 5 5 = 35
(5+5) : 5 5 = 10
Grading:
- speed – 1 b
- correctness - 2 b
- logic - 2 b
- Homework
Page 52 No. 6 (2) solve the problem, write the solution in the form of an expression.
- Result, reflection
Bloom's Cube
Name it topic of our lesson?
Explain the order of execution of actions in expressions with brackets.
Why Is it important to study this topic?
Continue first rule.
Come up with it algorithm for performing actions in expressions with brackets.
“If you want to participate in great life, then fill your head with mathematics while you have the opportunity. She will then be of great help to you in all your work.”(M.I. Kalinin)
Thanks for your work in class!!!
SHARE You canPrimary school is coming to an end, and soon the child will step into the advanced world of mathematics. But already during this period the student is faced with the difficulties of science. When performing a simple task, the child gets confused and lost, which ultimately leads to a negative mark for the work done. To avoid such troubles, when solving examples, you need to be able to navigate in the order in which you need to solve the example. Having distributed the actions incorrectly, the child does not complete the task correctly. The article reveals the basic rules for solving examples that contain the entire range of mathematical calculations, including brackets. Procedure in mathematics 4th grade rules and examples.
Before completing the task, ask your child to number the actions that he is going to perform. If you have any difficulties, please help.
Some rules to follow when solving examples without brackets:
If a task requires a number of actions to be performed, you must first perform division or multiplication, then . All actions are performed as the letter progresses. Otherwise, the result of the decision will not be correct.
If in the example you need to execute, we do it in order, from left to right.
27-5+15=37 (When solving the example, we are guided by the rule. First we perform subtraction, then addition).
Teach your child to always plan and number the actions performed.
The answers to each solved action are written above the example. This will make it much easier for the child to navigate the actions.
Let's consider another option where it is necessary to distribute actions in order:
As you can see, when solving, the rule is followed: first we look for the product, then we look for the difference.
This simple examples, when solving which, care is required. Many children are stunned when they see a task that contains not only multiplication and division, but also parentheses. A student who does not know the procedure for performing actions has questions that prevent him from completing the task.
As stated in the rule, first we find the product or quotient, and then everything else. But there are parentheses! What to do in this case?
Solving examples with brackets
Let's look at a specific example:
- When performing this task, we first find the value of the expression enclosed in parentheses.
- You should start with multiplication, then addition.
- After the expression in brackets is solved, we proceed to actions outside them.
- According to the rules of procedure, the next step is multiplication.
- The final stage will be.
As we see on clear example, all actions are numbered. To reinforce the topic, invite your child to solve several examples on their own:
The order in which the value of the expression should be calculated has already been arranged. The child will only have to carry out the decision directly.
Let's complicate the task. Let the child find the meaning of the expressions on his own.
7*3-5*4+(20-19) 14+2*3-(13-9)
17+2*5+(28-2) 5*3+15-(2-1*2)
24-3*2-(56-4*3) 14+12-3*(21-7)
Teach your child to solve all tasks in draft form. In this case, the student will have the opportunity to correct an incorrect decision or blots. Corrections are not allowed in the workbook. By completing tasks on their own, children see their mistakes.
Parents, in turn, should pay attention to mistakes, help the child understand and correct them. You shouldn’t overload a student’s brain with large amounts of tasks. With such actions you will discourage the child’s desire for knowledge. There should be a sense of proportion in everything.
Take a break. The child should be distracted and take a break from classes. The main thing to remember is that not everyone has a mathematical mind. Maybe your child will grow up to be a famous philosopher.
Order of actions - Mathematics 3rd grade (Moro)
Short description:
In life you constantly do various actions: get up, wash, do exercises, have breakfast, go to school. Do you think it is possible to change this procedure? For example, have breakfast and then wash your face. Probably possible. It may not be very convenient to have breakfast if you are unwashed, but nothing bad will happen because of this. In mathematics, is it possible to change the order of operations at your discretion? No, mathematics is an exact science, so even the slightest changes in the procedure will lead to the fact that the answer of the numerical expression will become incorrect. In second grade you have already become acquainted with some rules of procedure. So, you probably remember that the order in the execution of actions is governed by brackets. They show what actions need to be completed first. What other rules of procedure are there? Is the order of operations different in expressions with and without parentheses? 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 rules you have learned, 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 is especially important! 
This lesson discusses in detail the procedure for performing arithmetic operations in expressions without parentheses and with brackets. Students are given the opportunity, while completing assignments, to determine whether the meaning of expressions depends on the order in which arithmetic operations are performed, to find out whether the order of arithmetic operations is different in expressions without parentheses and with parentheses, to practice applying the learned rule, to find and correct errors made when determining the order of actions.
In life, we constantly perform some kind of action: we walk, study, read, write, count, smile, quarrel and make peace. We perform these actions in different orders. Sometimes they can be swapped, sometimes not. For example, when getting ready for school in the morning, you can first do exercises, then make your bed, or vice versa. But you can’t go to school first and then put on clothes.
In mathematics, is it necessary to perform arithmetic operations in a certain order?
Let's check
Let's compare the expressions:
8-3+4 and 8-3+4
We see that both expressions are exactly the same.
Let's perform actions in one expression from left to right, and in the other from right to left. You can use numbers to indicate the order of actions (Fig. 1).
Rice. 1. Procedure
In the first expression, we will first perform the subtraction operation and then add the number 4 to the result.
In the second expression, we first find the value of the sum, and then subtract the resulting result 7 from 8.
We see that the meanings of the expressions are different.
Let's conclude: The order in which arithmetic operations are performed cannot be changed.
Let's learn the rule for performing arithmetic operations in expressions without parentheses.
If an expression without parentheses includes only addition and subtraction or only multiplication and division, then the actions are performed in the order in which they are written.
Let's practice.
Consider the expression
This expression contains only addition and subtraction operations. These actions are called first stage actions.
We perform the actions from left to right in order (Fig. 2).
Rice. 2. Procedure
Consider the second expression
This expression contains only multiplication and division operations - These are the actions of the second stage.
We perform the actions from left to right in order (Fig. 3).
Rice. 3. Procedure
In what order are arithmetic operations performed if the expression contains not only addition and subtraction, but also multiplication and division?
If an expression without parentheses includes not only the operations of addition and subtraction, but also multiplication and division, or both of these operations, then first perform in order (from left to right) multiplication and division, and then addition and subtraction.
Let's look at the expression.
Let's think like this. This expression contains the operations of addition and subtraction, multiplication and division. We act according to the rule. First, we perform in order (from left to right) multiplication and division, and then addition and subtraction. Let's arrange the order of actions.
Let's calculate the value of the expression.
18:2-2*3+12:3=9-6+4=3+4=7
In what order are arithmetic operations performed if there are parentheses in an expression?
If an expression contains parentheses, the value of the expressions in the parentheses is evaluated first.
Let's look at the expression.
30 + 6 * (13 - 9)
We see that in this expression there is an action in parentheses, which means we will perform this action first, then multiplication and addition in order. Let's arrange the order of actions.
30 + 6 * (13 - 9)
Let's calculate the value of the expression.
30+6*(13-9)=30+6*4=30+24=54
How should one reason to correctly establish the order of arithmetic operations in a numerical expression?
Before starting calculations, you need to look at the expression (find out whether it contains parentheses, what actions it contains) and only then perform the actions in the following order:
1. actions written in brackets;
2. multiplication and division;
3. addition and subtraction.
The diagram will help you remember this simple rule (Fig. 4).
Rice. 4. Procedure
Let's practice.
Let's consider the expressions, establish the order of actions and perform calculations.
43 - (20 - 7) +15
32 + 9 * (19 - 16)
We will act according to the rule. The expression 43 - (20 - 7) +15 contains operations in parentheses, as well as addition and subtraction operations. Let's establish a procedure. The first action is to perform the operation in parentheses, and then, in order from left to right, subtraction and addition.
43 - (20 - 7) +15 =43 - 13 +15 = 30 + 15 = 45
The expression 32 + 9 * (19 - 16) contains operations in parentheses, as well as multiplication and addition operations. According to the rule, we first perform the action in parentheses, then multiplication (we multiply the number 9 by the result obtained by subtraction) and addition.
32 + 9 * (19 - 16) =32 + 9 * 3 = 32 + 27 = 59
In the expression 2*9-18:3 there are no parentheses, but there are multiplication, division and subtraction operations. We act according to the rule. First, we perform multiplication and division from left to right, and then subtract the result obtained from division from the result obtained by multiplication. That is, the first action is multiplication, the second is division, and the third is subtraction.
2*9-18:3=18-6=12
Let's find out whether the order of actions in the following expressions is correctly defined.
37 + 9 - 6: 2 * 3 =
18: (11 - 5) + 47=
7 * 3 - (16 + 4)=
Let's think like this.
37 + 9 - 6: 2 * 3 =
There are no parentheses in this expression, which means that we first perform multiplication or division from left to right, then addition or subtraction. In this expression, the first action is division, the second is multiplication. The third action should be addition, the fourth - subtraction. Conclusion: the procedure is determined correctly.
Let's find the value of this expression.
37+9-6:2*3 =37+9-3*3=37+9-9=46-9=37
Let's continue to talk.
The second expression contains parentheses, which means that we first perform the action in parentheses, then from left to right multiplication or division, addition or subtraction. We check: the first action is in parentheses, the second is division, the third is addition. Conclusion: the procedure is defined incorrectly. Let's correct the errors and find the meaning of the expression.
18:(11-5)+47=18:6+47=3+47=50
This expression also contains parentheses, which means that we first perform the action in parentheses, then from left to right multiplication or division, addition or subtraction. Let's check: the first action is in parentheses, the second is multiplication, the third is subtraction. Conclusion: the procedure is defined incorrectly. Let's correct the errors and find the meaning of the expression.
7*3-(16+4)=7*3-20=21-20=1
Let's complete the task.
Let's arrange the order of actions in the expression using the learned rule (Fig. 5).
Rice. 5. Procedure
We don't see numerical values, so we won't be able to find the meaning of expressions, but we'll practice applying the rule we've learned.
We act according to the algorithm.
The first expression contains parentheses, which means the first action is in parentheses. Then from left to right multiplication and division, then from left to right subtraction and addition.
The second expression also contains parentheses, which means we perform the first action in parentheses. After that, from left to right, multiplication and division, after that, subtraction.
Let's check ourselves (Fig. 6).
Rice. 6. Procedure
Today in class we learned about the rule for the order of actions in expressions without and with brackets.
Bibliography
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 1. - M.: “Enlightenment”, 2012.
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 2. - M.: “Enlightenment”, 2012.
- M.I. Moro. Math lessons: Guidelines for the teacher. 3rd grade. - M.: Education, 2012.
- Regulatory document. Monitoring and evaluation of learning outcomes. - M.: “Enlightenment”, 2011.
- "School of Russia": Programs for primary school. - M.: “Enlightenment”, 2011.
- S.I. Volkova. Mathematics: Test papers. 3rd grade. - M.: Education, 2012.
- V.N. Rudnitskaya. Tests. - M.: “Exam”, 2012.
- Festival.1september.ru ().
- Sosnovoborsk-soobchestva.ru ().
- Openclass.ru ().
Homework
1. Determine the order of actions in these expressions. Find the meaning of the expressions.
2. Determine in what expression this order of actions is performed:
1. multiplication; 2. division;. 3. addition; 4. subtraction; 5. addition. Find the meaning of this expression.
3. Make up three expressions in which the following order of actions is performed:
1. multiplication; 2. addition; 3. subtraction
1. addition; 2. subtraction; 3. addition
1. multiplication; 2. division; 3. addition
Find the meaning of these expressions.
