Math lessons: multiplication by zero is the main rule. The algorithm of the online calculator with examples
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.
Math-Calculator-Online v.1.0
The calculator performs the following operations: addition, subtraction, multiplication, division, working with decimals, extracting the root, raising to a power, calculating percentages, and other operations.
Solution:
How to use the math calculator
| Key | Designation | Explanation |
|---|---|---|
| 5 | numbers 0-9 | Arabic numerals. Enter natural integers, zero. To get a negative integer, press the +/- key |
| . | semicolon) | A decimal separator. If there is no digit before the dot (comma), the calculator will automatically substitute a zero before the dot. For example: .5 - 0.5 will be written |
| + | plus sign | Addition of numbers (whole, decimal fractions) |
| - | minus sign | Subtraction of numbers (whole, decimal fractions) |
| ÷ | division sign | Division of numbers (whole, decimal fractions) |
| X | multiplication sign | Multiplication of numbers (integers, decimals) |
| √ | root | Extracting the root from a number. When you press the "root" button again, the root is calculated from the result. For example: square root of 16 = 4; square root of 4 = 2 |
| x2 | squaring | Squaring a number. When you press the "squaring" button again, the result is squared. For example: square 2 = 4; square 4 = 16 |
| 1/x | fraction | Output to decimals. In the numerator 1, in the denominator the input number |
| % | percent | Get a percentage of a number. To work, you must enter: the number from which the percentage will be calculated, the sign (plus, minus, divide, multiply), how many percent in numerical form, the "%" button |
| ( | open bracket | An open parenthesis to set the evaluation priority. A closed parenthesis is required. Example: (2+3)*2=10 |
| ) | closed bracket | A closed parenthesis to set the evaluation priority. Mandatory open bracket |
| ± | plus minus | Changes sign to opposite |
| = | equals | Displays the result of the solution. Also, intermediate calculations and the result are displayed above the calculator in the "Solution" field. |
| ← | deleting a character | Deletes the last character |
| WITH | reset | Reset button. Completely resets the calculator to "0" |
The algorithm of the online calculator with examples
Addition.
Addition of whole natural numbers ( 5 + 7 = 12 )
Addition of whole natural and negative numbers ( 5 + (-2) = 3 )
Adding decimal fractional numbers ( 0.3 + 5.2 = 5.5 )
Subtraction.
Subtraction of whole natural numbers ( 7 - 5 = 2 )
Subtraction of whole natural and negative numbers ( 5 - (-2) = 7 )
Subtraction of decimal fractional numbers ( 6.5 - 1.2 = 4.3 )
Multiplication.
Product of whole natural numbers ( 3 * 7 = 21 )
Product of whole natural and negative numbers ( 5 * (-3) = -15 )
Product of decimal fractional numbers ( 0.5 * 0.6 = 0.3 )
Division.
Division of whole natural numbers ( 27 / 3 = 9 )
Division of whole natural and negative numbers ( 15 / (-3) = -5 )
Division of decimal fractional numbers ( 6.2 / 2 = 3.1 )
Extracting the root from a number.
Extracting the root of an integer ( root(9) = 3 )
Extracting the root of decimals ( root(2.5) = 1.58 )
Extracting the root from the sum of numbers ( root(56 + 25) = 9 )
Extracting the root of the difference in numbers ( root (32 - 7) = 5 )
Squaring a number.
Squaring an integer ( (3) 2 = 9 )
Squaring decimals ( (2.2) 2 = 4.84 )
Convert to decimal fractions.
Calculating percentages of a number
Increase 230 by 15% ( 230 + 230 * 0.15 = 264.5 )
Decrease the number 510 by 35% ( 510 - 510 * 0.35 = 331.5 )
18% of the number 140 is ( 140 * 0.18 = 25.2 )
For the first time with such an arithmetic operation as multiplication, students are introduced to school bench. The math teacher among the numerous rules raises the topic of "multiplying by zero." Despite the unambiguity of the wording, students have many questions. Let's look at what happens if we multiply by 0.
The rule that you cannot multiply by zero generates a lot of disputes between teachers and their students. It is important to understand that multiplication by zero is a controversial aspect due to its ambiguity.
First of all, attention is focused on the lack of a sufficient level of knowledge among secondary school students. secondary school. Crossing the threshold educational institution, a participant in the educational process in most cases does not think about the main goal that must be pursued.
During the training, the teacher covers various issues. These include the situation, what happens if you multiply by 0. In an effort to anticipate the teacher's narration, some students enter into controversy. They prove, at least they try, that multiplication by 0 is valid. But, unfortunately, this is not the case. Multiplying any number by 0 results in nothing. In some literary sources even there is a mention that any number multiplied by zero forms a void.
Important! Attentive audience listeners immediately grasp that if the number is multiplied by 0, then the result will be 0. A different development of events can be traced in the case of those students who systematically skip classes. Inattentive or unscrupulous students are more likely than others to think about how much it will be if they multiply by zero.
As a result of the lack of knowledge on the topic, the teacher and the negligent student find themselves on opposite sides of a contradictory situation.
The difference in views on the topic of the dispute lies in the degree of education on the subject of whether it is possible to multiply by 0 or still not. The only acceptable way out of this situation is to try to appeal to logical thinking to find the right answer.
It is not recommended to use the following example to explain the rule. Vanya has 2 apples in her bag for a snack. At lunch he thought about putting some more apples in his briefcase. But at that moment there was not a single fruit nearby. Vanya did not put anything. In other words, he placed 0 apples to 2 apples.
In terms of arithmetic this example it turns out that if 2 is multiplied by 0, then there is no emptiness. The answer in this case is clear. For this example, the multiplication by zero rule is not relevant. The right decision is summation. That is why the correct answer is 2 apples.
Otherwise, the teacher has no choice but to compose a series of tasks. The last measure is to re-set the passage of the topic and poll for exceptions in the multiplication.
Essence of action
It is advisable to start studying the algorithm of actions when multiplying by zero by indicating the essence of the arithmetic operation.
The essence of the action to multiply was originally determined exclusively for a natural number. If the mechanism of action is revealed, then a certain number involved in the calculation is added to itself.
It is important to consider the number of additions. Depending on this criterion, a different result is obtained. The addition of a number relative to itself determines such a property of it as naturalness.
Let's look at an example. It is necessary to multiply the number 15 by 3. When multiplied by 3, the number 15 increases three times in its value. In other words, the action looks like 15 * 3 = 15 + 15 + 15 = 45. Based on the calculation mechanism, it becomes obvious that if a number is multiplied by another natural number, there is a semblance of addition in a simplified form.
It is advisable to start the algorithm of actions when multiplying by 0 by providing a characteristic by zero.
Note! According to conventional wisdom, zero stands for the whole nothingness. For emptiness of this kind, a designation is provided in arithmetic. Despite given fact, a null value carries nothing.
It should be noted that such an opinion in the modern world scientific society differs from the point of view of the ancient Eastern scientists. According to the theory they held, zero was equal to infinity.
In other words, if you multiply by zero, you get a variety of options. In the zero value, scientists considered a kind of depth of the universe.
As confirmation of the possibility of multiplying by 0, mathematicians cited the following fact. If you put 0 next to any natural number, you get a value ten times greater than the original one.
The example given is one of the arguments. In addition to proofs of this kind, there are many other examples. It is they that underlie the ongoing disputes when multiplying by emptiness.
The feasibility of trying
Among students quite often at the beginning of mastering educational material there are attempts to multiply a number by 0. Such an action is a gross mistake.
In essence, nothing will happen from such attempts, but there will be no benefit either. If you multiply by a zero value, you get an unsatisfactory mark in the diary.
The only thought that should arise when multiplying by emptiness is the impossibility of action. memorization in this case plays an important role. Having learned the rule once and for all, the student prevents the appearance of controversial situations.
As an example to be used when multiplying by zero, the following situation is allowed to be used. Sasha decided to buy apples. While she was in the supermarket, she chose 5 large ripe apples. Going to the department of dairy products, she felt that this would not be enough for her. The girl put 5 more pieces in her basket.
After thinking a little more, she took 5 more. As a result, at the checkout, Sasha got: 5 * 3 = 5 + 5 + 5 = 15 apples. If she put 5 apples only 2 times, then it would be 5 * 2 = 5 + 5 = 10. In the event that Sasha did not put 5 apples in the basket, it would be 5 * 0 = 0 + 0 + 0 + 0 + 0 = 0. In other words, buying apples 0 times means not buying any.
Even at school, teachers tried to hammer the simplest rule into our heads: "Any number multiplied by zero equals zero!", - but still there is a lot of controversy 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.
In contact with
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.
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.
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.
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 otherwise - they approached the issue philosophically and drew some parallels between emptiness and infinity and saw 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 determine empty digits in decimal fractions, this is done both before and after the decimal point.
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, zero will still be obtained. 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 mysteries, as ancient scholars 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.
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. It will be clear even to a small 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.
Division
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 school curriculum, because there is not so much controversy and controversy 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. Therefore, the entry 10: 0 is the same abbreviation for 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!
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. Mathematics lessons: Guidelines for teachers. Grade 3 - M.: Education, 2012.
- Regulatory document. Monitoring and evaluation of learning outcomes. - M.: "Enlightenment", 2011.
- "School of Russia": Programs for elementary school. - M.: "Enlightenment", 2011.
- S.I. Volkov. Mathematics: Testing work. Grade 3 - M.: Education, 2012.
- V.N. Rudnitskaya. Tests. - M.: "Exam", 2012.
- 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.
