The number 0 can be represented as a kind of border separating the world of real numbers from imaginary or negative ones. Due to the ambiguous position, many operations with this numerical value do not obey mathematical logic. The impossibility of dividing by zero is a prime example of this. And allowed arithmetic operations with zero can be performed using generally accepted definitions.
History of Zero
Zero is the reference point in all standard number systems. Europeans began to use this number relatively recently, but the sages of ancient India used zero for a thousand years before the empty number was regularly used by European mathematicians. Even before the Indians, zero was a mandatory value in the Maya numerical system. This American people used the duodecimal system, and they began the first day of each month with a zero. Interestingly, among the Maya, the sign for "zero" completely coincided with the sign for "infinity". Thus, the ancient Maya concluded that these quantities were identical and unknowable.
Math operations with zero
Standard mathematical operations with zero can be reduced to a few rules.
Addition: if you add zero to an arbitrary number, then it will not change its value (0+x=x).
Subtraction: when subtracting zero from any number, the value of the subtracted remains unchanged (x-0=x).
Multiplication: any number multiplied by 0 gives 0 in the product (a*0=0).
Division: Zero can be divided by any non-zero number. In this case, the value of such a fraction will be 0. And division by zero is prohibited.
Exponentiation. This action can be performed with any number. An arbitrary number raised to the power of zero will give 1 (x 0 =1).
Zero to any power is equal to 0 (0 a \u003d 0).
In this case, a contradiction immediately arises: the expression 0 0 does not make sense.
Paradoxes of mathematics
The fact that division by zero is impossible, many people know from school. But for some reason it is not possible to explain the reason for such a ban. Indeed, why does the division-by-zero formula not exist, but other actions with this number are quite reasonable and possible? The answer to this question is given by mathematicians.
The thing is that the usual arithmetic operations that schoolchildren study in primary school are actually not as equal as we think. All simple operations with numbers can be reduced to two: addition and multiplication. These operations are the essence of the very concept of a number, and the rest of the operations are based on the use of these two.
Addition and multiplication
Let's take a standard subtraction example: 10-2=8. At school, it is considered simply: if two are taken away from ten objects, eight remain. But mathematicians look at this operation quite differently. After all, there is no such operation as subtraction for them. This example can be written in another way: x+2=10. For mathematicians, the unknown difference is simply the number that must be added to two to make eight. And no subtraction is required here, you just need to find a suitable numerical value.
Multiplication and division are treated in the same way. In the example of 12:4=3, it can be understood that we are talking about the division of eight objects into two equal piles. But in reality, this is just an inverted formula for writing 3x4 \u003d 12. Such examples for division can be given endlessly.
Examples for dividing by 0
This is where it becomes a little clear why it is impossible to divide by zero. Multiplication and division by zero have their own rules. All examples per division of this quantity can be formulated as 6:0=x. But this is an inverted expression of the expression 6 * x = 0. But, as you know, any number multiplied by 0 gives only 0 in the product. This property is inherent in the very concept of a zero value.
It turns out that such a number, which, when multiplied by 0, gives any tangible value, does not exist, that is given task has no solution. One should not be afraid of such an answer, it is a natural answer for problems of this type. Just writing 6:0 doesn't make any sense, and it can't explain anything. In short, this expression can be explained by the immortal "no division by zero".
Is there a 0:0 operation? Indeed, if the operation of multiplying by 0 is legal, can zero be divided by zero? After all, an equation of the form 0x5=0 is quite legal. Instead of the number 5, you can put 0, the product will not change from this.
Indeed, 0x0=0. But you still can't divide by 0. As said, division is just the inverse of multiplication. Thus, if in the example 0x5=0, you need to determine the second factor, we get 0x0=5. Or 10. Or infinity. Dividing infinity by zero - how do you like it?
But if any number fits into the expression, then it does not make sense, we cannot choose one from an infinite set of numbers. And if so, it means that the expression 0:0 does not make sense. It turns out that even zero itself cannot be divided by zero.
higher mathematics
Division by zero is headache for school mathematics. Studied in technical universities mathematical analysis slightly expands the concept of problems that have no solution. For example, to already famous expression 0:0 new ones are added that have no solution in school courses mathematics:
- infinity divided by infinity: ∞:∞;
- infinity minus infinity: ∞−∞;
- unit raised to an infinite power: 1 ∞ ;
- infinity multiplied by 0: ∞*0;
- some others.
It is impossible to solve such expressions by elementary methods. But higher mathematics, thanks to additional possibilities for a number of similar examples, gives final solutions. This is especially evident in the consideration of problems from the theory of limits.
Uncertainty Disclosure
In the theory of limits, the value 0 is replaced by the conditional infinitesimal variable. And expressions in which division by zero is obtained when substituting the desired value are converted. Below is a standard example of limit expansion using the usual algebraic transformations:
As you can see in the example, a simple reduction of a fraction brings its value to a completely rational answer.
When considering the limits trigonometric functions their expressions tend to be reduced to the first wonderful limit. When considering the limits in which the denominator goes to 0 when the limit is substituted, the second remarkable limit is used.
L'Hopital Method
In some cases, the limits of expressions can be replaced by the limit of their derivatives. Guillaume Lopital - French mathematician, founder of the French school mathematical analysis. He proved that the limits of expressions are equal to the limits of the derivatives of these expressions. In mathematical notation, his rule is as follows.
Presentation for the lesson
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- Introduce special cases of multiplication with 0 and 1.
- To consolidate the meaning of multiplication and the commutative property of multiplication, to develop computational skills.
- Develop attention, memory, mental operations, speech, creativity, interest in mathematics.
Equipment: Slide presentation: Appendix1.
1. Organizational moment.
Today is an unusual day for us. There are guests at the lesson. Please me, friends, guests with your successes. Open notebooks, write down the number, class work. In the margin, mark your mood at the beginning of the lesson. Slide 2.
Verbally the whole class repeats the multiplication table on the cards with speaking aloud (Children mark wrong answers with claps).
Fizkultminutka (“Brain gymnastics”, “Hat for reflection”, for breathing).
2. Statement of the learning task.
2.1. Tasks for the development of attention.
On the board and on the table, the children have a two-color picture with numbers:
– What is interesting about the written numbers? (Written in different colors; all “red” numbers are even, and “blue” are odd.)
What is the extra number? (10 is round and the rest are not; 10 is two digits and the rest are single digits; 5 is repeated twice and the rest are one at a time.)
- I will close the number 10. Is there an extra among the other numbers? (3 - he doesn't have a pair under 10, but the others do.)
– Find the sum of all “red” numbers and write it down in the red square. (30.)
– Find the sum of all the “blue” numbers and write it down in the blue square. (23.)
How much more is 30 than 23? (On 7.)
How much is 23 less than 30? (Also at 7.)
What action were you looking for? (Subtraction.) Slide 3.
2.2. Tasks for the development of memory and speech. Knowledge update.
a) - Repeat in order the words that I will name: term, term, sum, reduced, subtracted, difference. (Children try to reproduce word order.)
What action components were named? (Addition and subtraction.)
What action are you familiar with? (Multiplication, division.)
- Name the components of multiplication. (Multiplier, multiplier, product.)
What does the first multiplier mean? (Equal terms in the sum.)
What does the second multiplier mean? (The number of such terms.)
Write down the definition of multiplication.
b) Look at the notes. What task will you be doing?
12 + 12 + 12 + 12 + 12
33 + 33 + 33 + 33
a + a + a
(Replace sum by product.)
What will happen? (The first expression has 5 terms, each of which is equal to 12, so it is equal to 12 5. Similarly - 33 4, and 3)
c) Name the reverse operation. (Replace the product with the sum.)
– Replace the product with the sum in the expressions: 99 2. 8 4. b 3. (99 + 99, 8 + 8 + 8 + 8, b + b + b). slide 4.
d) Equations are written on the board:
81 + 81 = 81 – 2
21 3 = 21 + 22 + 23
44 + 44 + 44 + 44 = 44 + 4
17 + 17 – 17 + 17 – 17 = 17 5
Pictures are placed next to each equality.
The animals of the forest school were on a mission. Did they do it right?
Children establish that the elephant, tiger, hare and squirrel made a mistake, explain what their mistakes are. Slide 5.
e) Compare the expressions:
8 5. 5 8
5 6. 3 6
34 9… 31 2
a 3. a 2 + a
(8 5 \u003d 5 8, since the sum does not change from the rearrangement of the terms;
5 6 > 3 6, since there are 6 terms on the left and on the right, but the terms on the left are larger;
34 9 > 31 2. since there are more terms on the left and the terms themselves are larger;
a 3 \u003d a 2 + a, since there are 3 terms on the left and on the right, equal to a.)
What property of multiplication was used in the first example? (Displacement.) Slide 6.
2.3. Formulation of the problem. Goal setting.
Are equalities true? Why? (Correct, since the sum is 5 + 5 + 5 = 15. Then the sum becomes 5 more by one term, and the sum increases by 5.)
5 3 = 15
5 4 = 20
5 5 = 25
5 6 = 30
– Continue this pattern to the right. (5 7 = 35; 5 8 = 40.)
- Continue it now to the left. (5 2 = 10; 5 1=5; 5 0 = 0.)
- What does the expression 5 1 mean? 50? (? Problem!)
However, the expressions 5 1 and 5 0 do not make sense. We can agree to consider these equalities true. But for this we need to check whether we violate the commutative property of multiplication.
So, the purpose of our lesson is determine if we can count the equalities 5 1 = 5 and 5 0 = 0 correct?
Lesson problem! Slide 7.
3. “Discovery” of new knowledge by children.
a) - Follow the steps: 1 7, 1 4, 1 5.
Children solve examples with comments in a notebook and on the board:
1 7 = 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7
1 4 = 1 + 1 + 1 + 1 = 4
1 5 = 1 + 1 + 1 + 1 +1 = 5
- Make a conclusion: 1 a -? (1 a = a.) Card is exposed: 1 a = a
b) - Do the expressions 7 1, 4 1, 5 1 make sense? Why? (No, since the sum cannot have one term.)
– What should they be equal to in order not to violate the commutative property of multiplication? (7 1 must also equal 7, so 7 1 = 7.)
4 1 = 4; 5 1 = 5.
- Make a conclusion: a 1 =? (a 1 = a.)
The card is exposed: a 1 = a. The first card is superimposed on the second: a 1 \u003d 1 a \u003d a.
- Does our conclusion coincide with what we got on the numerical beam? (Yes.)
– Translate this equality into Russian. (When you multiply a number by 1 or 1 by a number, you get the same number.)
- Well done! So, we will consider: a 1 \u003d 1 a \u003d a. slide 8.
2) The case of multiplication with 0 is studied similarly. Conclusion:
- when a number is multiplied by 0 or 0 by a number, zero is obtained: a 0 \u003d 0 a \u003d 0. slide 9.
- Compare both equalities: what do 0 and 1 remind you of?
Children express their opinions. You can draw their attention to the images:
1 - “mirror”, 0 - “terrible beast” or “invisibility cap”.
Well done! So, multiplying by 1 gives the same number. (1 - “mirror”), and when multiplied by 0, we get 0 ( 0 - “invisibility cap”).
4. Physical education (for the eyes - “circle”, “up - down”, for hands - “lock”, “cams”).
5. Primary fastening.
Examples are written on the board:
Children solve them in a notebook and on a blackboard with pronunciation of the received rules in a loud speech, for example:
3 1 = 3, since when multiplying a number by 1, the same number is obtained (1 is a “mirror”), etc.
a) 145 x = 145; b) x 437 = 437.
- When multiplying 145 by an unknown number, it turned out 145. So, they multiplied by 1 x = 1. Etc.
- Multiplying 8 by an unknown number turned out to be 0. So, multiplied by 0 x \u003d 0. And so on.
6. Independent work with class validation. slide 10.
Children independently solve recorded examples. Then finished
they check their answers with pronunciation in a loud speech, mark correctly solved examples with a plus, correct the mistakes made. Those who made mistakes receive a similar task on a card and work on it individually while the class solves repetition problems.
7. Tasks for repetition. (Work in pairs). Slide 11.
a) - Do you want to know what awaits you in the future? You can find out by deciphering the record:
xn--i1abbnckbmcl9fb.xn--p1ai
Multiplication by 1 and 0 rule
According to the generally accepted definition, zero is the number that separates positive numbers from negative numbers on the number line. Zero- this is the most problematic place in mathematics, which does not obey logic, and all mathematical operations with zero based not on logic, but on generally accepted definitions.
The first example of problematic zero are natural numbers. in Russian schools zero is not natural number, in other schools zero is a natural number. Since the concept of “natural numbers” is an artificial separation of some numbers from all other numbers according to certain criteria, there can be no mathematical proof of the naturalness or non-naturalness of zero. Zero is considered a neutral element with respect to addition and subtraction operations.
Zero is considered an integer, unsigned number. Also zero is considered an even number, because when you divide zero by 2, you get an integer zero.
Zero is the first digit in all standard number systems. In positional number systems, to which the decimal number system familiar to us belongs, the digit zero indicate the absence of a value for this bit when writing a number. The Maya Indians used zero in their duodecimal number system a thousand years before Indian mathematicians. Every month began from day zero in the Mayan calendar. Interestingly, the same sign zero Mayan mathematicians also denoted infinity - the second problem of modern mathematics.
Word " zero" V Arabic sounds like "syfr". From the Arabic word zero(syfr) the word "number" occurred.
How to spell - zero or zero? The words zero and zero have the same meaning, but differ in usage. Usually, zero used in everyday speech and in a number of stable combinations, zero- in terminology, in scientific speech. Both spellings of this word are correct. For example: Division by zero. Zero whole. Zero attention. Zero without a wand. absolute zero. Zero point five.
In grammar, derivative words from words zero And zero are written like this: zero or zero, zero or zero, zero or zero, zero or less common zero, zero-zero. For example: Below zero. Equals zero. Reduce to zero. Zero meridian. Zero mileage. At twelve zero zero.
IN mathematical operations with zero to date, the following results have been defined:
addition- if you add to any number zero, the number will remain unchanged; if to zero add any number, the result of addition will be the same any number:
subtraction- if you subtract from any number zero, the number will remain unchanged; if from zero subtract any number, the result will be the same any number with the opposite sign:
multiplication- if any number is multiplied by zero, the result is zero; If zero is multiplied by any number, the result is zero:
division- division by zero forbidden because the result does not exist; the generally accepted view of the problem of division by zero is set out in the work of Alexander Sergeev " Why can't you divide by zero?» ; for the curious, another article has been written that discusses the possibility of dividing by zero:
a: 0 = no divide by zero, wherein A not equal to zero
zero divide by zero- the expression does not make sense, since it cannot be defined:
0: 0 = expression does not make sense
zero divided by a number- If zero divided by a number the result will always be zero, no matter what number is in the denominator (an exception to this rule is the number zero, see above):
0:a=0, wherein A not equal to zero
zero to the power — zero equal to any extent zero:
0 a = 0, wherein A not equal to zero
exponentiation- any number to the power zero equals one (number to the power of 0):
a 0 = 1, wherein A not equal to zero
zero to the power of zero- the expression does not make sense, since it cannot be defined (zero to the zero power, 0 to the power of 0):
0 0 = expression does not make sense
root extraction is any degree root of zero equals zero:
0 1/a = 0, wherein A not equal to zero
factorial- factorial of zero, or zero factorial, equals one:
digit distribution- when calculating the distribution of numbers zero considered an insignificant number. Changing the approach in the rules for counting the distribution of digits when zero considered a SIGNIFICANT digit will allow you to get more accurate results of the distribution of digits in all standard number systems, including the binary number system.
Who is interested in the question of zero, I propose to read the article "The History of Zero" by J. J. O'Connor and E. F. Robertson, translated by I. Yu. Osmolovsky.
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How to multiply by 0.1
Let's analyze the rule and look at examples of how to multiply any number by 0.1.
Therefore, multiplying a number by 0.1 can be replaced by dividing it by 10. In general view it can be written like this:
This is where the rule comes in.
0.1 multiplication rule
To multiply a number by 0.1, you need to move the comma in the record of this number one digit to the left.
When writing a natural number, do not write a comma at the end:
Multiplying a natural number by 0.1 means moving this comma one character to the left:
If the last digit in the record of a natural number is zero, as a result of multiplying this number by 0.1, we get a natural number (since zero after the decimal point at the end of the number is not written):
To multiply by 0.1 common fraction, it is necessary to bring both fractions to the same form - either convert the ordinary fraction to decimal, or decimal to ordinary.
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Rule for multiplying any number by zero
Even at school, teachers tried to hammer the simplest rule into our heads: "Any number multiplied by zero equals zero!", - but still a lot of controversy constantly arises around him. Someone just memorized the rule and does not bother with the question “why?”. “You can’t do everything here, because at school they said so, the rule is the rule!” Someone can fill half a notebook with formulas, proving this rule or, conversely, its illogicality.
Who is right in the end
During these disputes, both people, having opposite points of view, look at each other like a ram, and prove with all their might that they are right. Although, if you look at them from the side, you can see not one, but two rams resting against each other with their horns. The only difference between them is that one is slightly less educated than the other.
This is interesting: bit terms - what is it?
Most often, those who consider this rule to be wrong try to call for logic in this way:
I have two apples on my table, if I put zero apples to them, that is, I don’t put a single one, then my two apples will not disappear from this! The rule is illogical!
Indeed, apples will not disappear anywhere, but not because the rule is illogical, but because a slightly different equation is used here: 2 + 0 \u003d 2. So we will immediately discard such a conclusion - it is illogical, although it has the opposite goal - to call to logic.
This is interesting: How to find the difference of numbers in mathematics?
What is multiplication
The original multiplication rule was defined only for natural numbers: multiplication is a number added to itself a certain number of times, which implies the naturalness of the number. Thus, any number with multiplication can be reduced to this equation:
- 25x3=75
- 25 + 25 + 25 = 75
- 25x3 = 25 + 25 + 25
From this equation follows the conclusion, that multiplication is a simplified addition.
This is interesting: what is a circle chord in geometry, definition and properties.
What is zero
Any person knows from childhood: zero is emptiness. Despite the fact that this emptiness has a designation, it does not carry anything at all. Ancient Eastern scientists thought differently - they approached the issue philosophically and drew some parallels between emptiness and infinity and saw a deep meaning in this number. After all, zero, which has the value of emptiness, standing next to any natural number, multiplies it ten times. Hence all the controversy over multiplication - this number carries so much inconsistency that it becomes difficult not to get confused. In addition, zero is constantly used to identify empty bits in decimal fractions, this is done both before and after the comma.
Is it possible to multiply by emptiness
It is possible to multiply by zero, but it is useless, because, whatever one may say, but even when multiplying negative numbers it will still be zero. It is enough just to remember this simplest rule and never ask this question again. In fact, everything is simpler than it seems at first glance. There are no hidden meanings and secrets, as ancient scientists believed. The most logical explanation will be given below that this multiplication is useless, because when multiplying a number by it, the same thing will still be obtained - zero.
This is interesting: what is the modulus of a number?
Going back to the very beginning, the argument about two apples, 2 times 0 looks like this:
- If you eat two apples five times, then eaten 2×5 = 2+2+2+2+2 = 10 apples
- If you eat two of them three times, then eaten 2 × 3 = 2 + 2 + 2 = 6 apples
- If you eat two apples zero times, then nothing will be eaten - 2x0 = 0x2 = 0+0 = 0
After all, eating an apple 0 times means not eating a single one. This will be clear even to the smallest child. Like it or not, 0 will come out, two or three can be replaced with absolutely any number and absolutely the same thing will come out. And to put it simply, zero is nothing and when you have there is nothing, then no matter how much you multiply - it's all the same will be zero. There is no magic, and nothing will make an apple, even if you multiply 0 by a million. This is the simplest, most understandable and logical explanation of the rule of multiplication by zero. For a person who is far from all formulas and mathematics, such an explanation will be enough for the dissonance in the head to resolve and everything to fall into place.
From all of the above follows another important rule:
You can't divide by zero!
This rule, too, has been stubbornly hammered into our heads since childhood. We just know that it is impossible and that's it, without stuffing our heads with unnecessary information. If you are suddenly asked the question, for what reason it is forbidden to divide by zero, then the majority will be confused and will not be able to clearly answer the simplest question from the school curriculum, because there are not so many disputes and contradictions around this rule.
Everyone just memorized the rule and does not divide by zero, not suspecting that the answer lies on the surface. Addition, multiplication, division and subtraction are unequal, only multiplication and addition are full of the above, and all other manipulations with numbers are built from them. That is, the entry 10: 2 is an abbreviation of the equation 2 * x = 10. Hence, the entry 10: 0 is the same abbreviation of 0 * x = 10. It turns out that division by zero is a task to find a number, multiplying by 0, you get 10 And we have already figured out that such a number does not exist, which means that this equation has no solution, and it will be a priori incorrect.
Let me tell you
To not divide by 0!
Cut 1 as you like, along,
Just don't divide by 0!
obrazovanie.guru
Multiplication with 0 and 1. 2nd grade
Presentation for the lesson
Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested this work please download the full version.
Lesson Objectives:
- Educational:
- to form the ability to perform multiplication with zero and one;
- to form the ability to correctly read mathematical expressions, name the components of multiplication;
- to consolidate the ability to replace the product of numbers with the sum and verbally calculate their value; to form the initial skills of working with the test.
- Educational:
- to promote the development of mathematical speech, working memory, voluntary attention, visual-effective thinking.
- Educational:
- develop a culture of behavior front work, individual work; interest in the subject.
Lesson type- a lesson in the discovery of new knowledge.
The formation of new skills is possible only in activity, therefore, in the development of the lesson, the technology of the activity method was used. The use of this technology is a significant factor in increasing the efficiency of students mastering subject knowledge, the formation of educational universal action: regulatory, communicative, cognitive.
The developed lesson has the following structure:
1. Acquisition of primary experience in performing an action and motivation.
2. Formation of a new method (algorithm) of action, establishment of primary links with existing methods.
3. Training, clarification of connections, self-control and correction.
4. Control.
Equipment for the lesson:
- Standard: a textbook, a table for filling out test answers, colored paper stars, memos for students.
- Innovative: multimedia projector, interactive whiteboard, multimedia presentation "Journey to the Planet of Multiplication"
The use of multimedia components in the lesson introduces an element of novelty, makes the work process visual, and helps the teacher to focus on the main points. Work on each stage of the lesson is built as a kind of dialogue between the teacher and students, in which the interactive whiteboard serves as a demonstrator for solving questions. Its use in educational process allows you to achieve high degree effectiveness.
Class: 3
Presentation for the lesson
Back forward
Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested in this work, please download the full version.
Target:
- Introduce special cases of multiplication with 0 and 1.
- To consolidate the meaning of multiplication and the commutative property of multiplication, to develop computational skills.
- Develop attention, memory, mental operations, speech, creativity, interest in mathematics.
Equipment: Slide presentation: Appendix1.
During the classes
1. Organizational moment.
Today is an unusual day for us. There are guests at the lesson. Please me, friends, guests with your successes. Open notebooks, write down the number, class work. In the margin, mark your mood at the beginning of the lesson. Slide 2.
Verbally the whole class repeats the multiplication table on the cards with speaking aloud (Children mark wrong answers with claps).
Fizkultminutka (“Brain gymnastics”, “Hat for reflection”, for breathing).
2. Statement of the learning task.
2.1. Tasks for the development of attention.
On the board and on the table, the children have a two-color picture with numbers:
– What is interesting about the written numbers? (Written in different colors; all “red” numbers are even, and “blue” are odd.)
What is the extra number? (10 is round and the rest are not; 10 is two digits and the rest are single digits; 5 is repeated twice and the rest are one at a time.)
- I will close the number 10. Is there an extra among the other numbers? (3 - he doesn't have a pair under 10, but the others do.)
– Find the sum of all “red” numbers and write it down in the red square.
(30.)
– Find the sum of all the “blue” numbers and write it down in the blue square. (23.)
How much more is 30 than 23? (On 7.)
How much is 23 less than 30? (Also at 7.)
What action were you looking for? (Subtraction.) Slide 3.
2.2. Tasks for the development of memory and speech. Knowledge update.
a) - Repeat in order the words that I will name: term, term, sum, reduced, subtracted, difference. (Children try to reproduce word order.)
What action components were named? (Addition and subtraction.)
What action are you familiar with? (Multiplication, division.)
- Name the components of multiplication. (Multiplier, multiplier, product.)
What does the first multiplier mean? (Equal terms in the sum.)
What does the second multiplier mean? (The number of such terms.)
Write down the definition of multiplication.
a + a+… + a= an
b) Look at the notes. What task will you be doing?
12 + 12 + 12 + 12 + 12
33 + 33 + 33 + 33
a + a + a
(Replace sum by product.)
What will happen? (The first expression has 5 terms, each of which is equal to 12, so it is equal to 12 5. Similarly - 33 4, and 3)
c) Name the reverse operation. (Replace the product with the sum.)
– Replace the product with the sum in the expressions: 99 2. 8 4. b 3.(99 + 99, 8 + 8 + 8 + 8, b + b + b). slide 4.
d) Equations are written on the board:
81 + 81 = 81 – 2
21 3 = 21 + 22 + 23
44 + 44 + 44 + 44 = 44 + 4
17 + 17 – 17 + 17 – 17 = 17 5
Pictures are placed next to each equality.
The animals of the forest school were on a mission. Did they do it right?
Children establish that the elephant, tiger, hare and squirrel made a mistake, explain what their mistakes are. Slide 5.
e) Compare the expressions:
8 5... 5 8
5 6... 3 6
34 9… 31 2
a 3... a 2 + a
(8 5 \u003d 5 8, since the sum does not change from the rearrangement of the terms;
5 6 > 3 6, since there are 6 terms on the left and on the right, but the terms on the left are larger;
34 9 > 31 2. since there are more terms on the left and the terms themselves are larger;
a 3 \u003d a 2 + a, since there are 3 terms on the left and on the right, equal to a.)
What property of multiplication was used in the first example? (Displacement.) Slide 6.
2.3. Formulation of the problem. Goal setting.
Are equalities true? Why? (Correct, since the sum is 5 + 5 + 5 = 15. Then the sum becomes 5 more by one term, and the sum increases by 5.)
5 3 = 15
5 4 = 20
5 5 = 25
5 6 = 30
– Continue this pattern to the right. (5 7 = 35; 5 8 = 40...)
- Continue it now to the left. (5 2 = 10; 5 1=5; 5 0 = 0.)
- What does the expression 5 1 mean? 50? (? Problem!)
Outcome of the discussion:
However, the expressions 5 1 and 5 0 do not make sense. We can agree to consider these equalities true. But for this we need to check whether we violate the commutative property of multiplication.
So, the purpose of our lesson is determine if we can count the equalities 5 1 = 5 and 5 0 = 0 correct?
Lesson problem! Slide 7.
3. “Discovery” of new knowledge by children.
a) - Follow the steps: 1 7, 1 4, 1 5.
Children solve examples with comments in a notebook and on the board:
1 7 = 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7
1 4 = 1 + 1 + 1 + 1 = 4
1 5 = 1 + 1 + 1 + 1 +1 = 5
- Make a conclusion: 1 a -? (1 a = a.) Card is exposed: 1 a = a
b) - Do the expressions 7 1, 4 1, 5 1 make sense? Why? (No, since the sum cannot have one term.)
– What should they be equal to in order not to violate the commutative property of multiplication? (7 1 must also equal 7, so 7 1 = 7.)
4 1 = 4; 5 1 = 5.
- Make a conclusion: a 1 =? (a 1 = a.)
The card is exposed: a 1 = a. The first card is superimposed on the second: a 1 \u003d 1 a \u003d a.
- Does our conclusion coincide with what we got on the numerical beam? (Yes.)
– Translate this equality into Russian. (When you multiply a number by 1 or 1 by a number, you get the same number.)
- Well done! So, we will consider: a 1 \u003d 1 a \u003d a. slide 8.
2) The case of multiplication with 0 is studied similarly. Conclusion:
- when a number is multiplied by 0 or 0 by a number, zero is obtained: a 0 \u003d 0 a \u003d 0. slide 9.
- Compare both equalities: what do 0 and 1 remind you of?
Children express their opinions. You can draw their attention to the images:
1 - “mirror”, 0 - “terrible beast” or “invisibility cap”.
Well done! So, multiplying by 1 gives the same number. (1 - “mirror”), and when multiplied by 0, we get 0 ( 0 - “invisibility cap”).
4. Physical education (for the eyes - “circle”, “up - down”, for hands - “lock”, “cams”).
5. Primary fastening.
Examples are written on the board:
23 1 =
1 89 =
0 925 =
364 1 =
156 0 =
0 1 =
Children solve them in a notebook and on a blackboard with pronunciation of the received rules in a loud speech, for example:
3 1 = 3, since when multiplying a number by 1, the same number is obtained (1 is a “mirror”), etc.
a) 145 x = 145; b) x 437 = 437.
- When multiplying 145 by an unknown number, it turned out 145. So, they multiplied by 1 x = 1. Etc.
a) 8 x = 0; b) x 1 \u003d 0.
- Multiplying 8 by an unknown number turned out to be 0. So, multiplied by 0 x \u003d 0. And so on.
6. Independent work with class check. slide 10.
Children independently solve recorded examples. Then finished
they check their answers with pronunciation in a loud speech, mark correctly solved examples with a plus, correct the mistakes made. Those who made mistakes receive a similar task on a card and work on it individually while the class solves repetition problems.
7. Tasks for repetition. (Work in pairs). Slide 11.
a) - Do you want to know what awaits you in the future? You can find out by deciphering the record:
G – 49:7 O – 9 8 n – 9 9 V – 45:5 th – 6 6 d – 7 8 s – 24:3
| 81 | 72 | 5 | 8 | 36 | 7 | 72 | 56 |
"So what's in store for us?" (New Year.)
b) - “I thought of a number, subtracted 7 from it, added 15, then added 4 and got 45. What number did I think of?”
Reverse operations must be done in reverse order: 45 - 4 - 15 + 7 = 31.
8. The result of the lesson.slide 12.
What are the new rules?
What did you like? What was difficult?
Can this knowledge be applied in real life?
In the margins, you can express your mood at the end of the lesson.
Complete the self-assessment table:
I want to know more
ok but i can do better
While I'm in trouble
Thanks for your work, you did a great job!
9. Homework
pp. 72–73 Rule, No. 6.
If we can rely on other laws of arithmetic, then this particular fact can be proved.
Suppose there is a number x for which x * 0 = x", and x" is not zero (for simplicity, we will assume that x" > 0)
Then, on the one hand, x * 0 = x", on the other hand, x * 0 = x * (1 - 1) = x - x
It turns out that x - x = x", whence x = x + x", i.e. x > x, which cannot be true.
This means that our assumption leads to a contradiction and there is no such number x for which x * 0 would not be equal to zero.
the assumption cannot be true because it is just an assumption! nobody plain language can't explain or find it difficult! if 0 * x = 0 then 0 * x = (0 + 0) * x \u003d 0 * x + 0 * x and as a result they reduced the right to the left 0 \u003d 0 * x this is supposedly a mathematical proof! but such nonsense with this zero terribly contradicts and in my opinion 0 should not be a number, but only an abstract concept! So that mere mortals would not be burned in the brain by the fact that the physical presence of objects, when miraculously multiplied by nothing, gave rise to nothing!
P / s it’s not entirely clear to me, not a mathematician, but to a mere mortal where did you get units in the reasoning equation (like 0 is the same as 1-1)
I'm crazy about reasoning like there is some kind of X and let it be any number
is in the equation 0 and when multiplied by it, we set all numerical values to zero
therefore X is a numeric value, and 0 is the number of actions performed on the number X (and the actions, in turn, are also displayed in a numeric format)
EXAMPLE on apples)) :
Kolya had 5 apples, he took these apples and went to the market in order to increase capital, but the day turned out to be rainy, cloudy trade did not work out and Kalek returned home with nothing. In mathematical language, the story about Kolya and apples looks like this
5 apples * 0 sales = made 0 profits 5*0=0
Before going to the bazaar, Kolya went and picked 5 apples from a tree, and tomorrow he went to pick but did not reach for some reason of his own ...
Apples 5, tree 1, 5*1=5 (Kolya picked 5 apples on the 1st day)
Apples 0, tree 1, 0*1=0 (actually the result of Kolya's work on the second day)
The scourge of mathematics is the word "Suppose"
Answer
And if in another way, 5 apples for 0 apples \u003d how many apples, in mathematics it should be zero, and so
In fact, any numbers make sense only when they are associated with material objects, such as 1 cow, 2 cows, or whatever, and an account has appeared in order to count objects and not just like that, and there is a paradox if I don’t have a cow , and the neighbor has a cow, and we multiply my absence by the neighbor’s cow, then his cow should disappear, multiplication is generally invented to facilitate addition large quantities identical items, when it is difficult to count them using the addition method, for example, money was stacked in columns of 10 coins, and then the number of columns was multiplied by the number of coins in the column, much easier than adding. but if the number of columns is multiplied by zero coins, then it will naturally turn out to be zero, but if there are both columns and coins, then how not to multiply them by zero, the coins will not go anywhere because they are, and even if it is one coin, then the column is consisting of one coin, so you can’t get anywhere, so zero when multiplied by zero is obtained only under certain conditions, that is, in the absence of a material component, and if I have 2 socks, since you don’t multiply them by zero, they won’t go anywhere .
Which of these sums do you think can be replaced by the product?
Let's argue like this. In the first sum, the terms are the same, the number five is repeated four times. So we can replace addition with multiplication. The first factor shows which term is repeated, the second factor shows how many times this term is repeated. We replace the sum with the product.
Let's write down the solution.
In the second sum, the terms are different, so it cannot be replaced by a product. We add the terms and get the answer 17.
Let's write down the solution.
Can the product be replaced by the sum of the same terms?
Consider works.
Let's take action and draw a conclusion.
1*2=1+1=2
1*4=1+1+1+1=4
1*5=1+1+1+1+1=5
We can conclude: always the number of unit terms is equal to the number by which the unit is multiplied.
Means, multiplying the number one by any number gives the same number.
1 * a = a
Consider works.
These products cannot be replaced by a sum, since the sum cannot have one term.
The products in the second column differ from the products in the first column only in the order of the factors.
This means that in order not to violate the commutative property of multiplication, their values must also be equal, respectively, to the first factor.
Let's conclude: When any number is multiplied by the number one, the number that was multiplied is obtained.
We write this conclusion as an equality.
a * 1= a
Solve examples.
Hint: do not forget the conclusions that we made in the lesson.
Test yourself.
Now let's observe the products, where one of the factors is zero.
Consider products where the first factor is zero.
Let us replace the products with the sum of identical terms. Let's take action and draw a conclusion.
0*3=0+0+0=0
0*6=0+0+0+0+0+0=0
0*4=0+0+0+0=0
The number of zero terms is always equal to the number by which zero is multiplied.
Means, When you multiply zero by a number, you get zero.
We write this conclusion as an equality.
0 * a = 0
Consider products where the second factor is zero.
These products cannot be replaced by a sum, since the sum cannot have zero terms.
Let's compare the works and their meanings.
0*4=0
The products of the second column differ from the products of the first column only in the order of the factors.
This means that in order not to violate the commutative property of multiplication, their values must also be equal to zero.
Let's conclude: Multiplying any number by zero results in zero.
We write this conclusion as an equality.
a * 0 = 0
But you can't divide by zero.
Solve examples.
Hint: don't forget the conclusions drawn in the lesson. When calculating the values of the second column, be careful when determining the order of operations.
Test yourself.
Today in the lesson we got acquainted with special cases of multiplication by 0 and 1, practiced multiplying by 0 and 1.
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. Mathematics: Verification work. Grade 3 - M.: Education, 2012.
- V.N. Rudnitskaya. Tests. - M.: "Exam", 2012.
- Nsportal.ru ().
- Prosv.ru ().
- Do.gendocs.ru ().
Homework
1. Find the meaning of expressions.
2. Find the meaning of expressions.
3. Compare expression values.
(56-54)*1 … (78-70)*1
4. Make a task on the topic of the lesson for your comrades.
