The division of natural numbers, especially multi-digit ones, is conveniently carried out by a special method, which is called division by a column (in a column). You can also find the name corner division. Let us immediately note that the column can be used to both divide natural numbers without a remainder and divide natural numbers with a remainder.

In this article we will look at how long division is performed. Here we will talk about recording rules and all intermediate calculations. First, let's focus on dividing a multi-digit natural number by a single-digit number with a column. After this, we will focus on cases when both the dividend and the divisor are multi-valued natural numbers. The entire theory of this article is provided with typical examples of division by a column of natural numbers with detailed explanations of the solution and illustrations.

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Rules for recording when dividing by a column

Let's start by studying the rules for writing the dividend, divisor, all intermediate calculations and results when dividing natural numbers by a column. Let’s say right away that it is most convenient to do column division in writing on paper with a checkered line - this way there is less chance of straying from the desired row and column.

First, the dividend and divisor are written in one line from left to right, after which a symbol of the form is displayed between the written numbers. For example, if the dividend is the number 6 105 and the divisor is 5 5, then their correct entry when dividing into a column will be as follows:

Look at the following diagram to illustrate where to write the dividend, divisor, quotient, remainder, and intermediate calculations in long division.

From the above diagram it is clear that the required quotient (or incomplete quotient when dividing with a remainder) will be written below the divisor under the horizontal line. And intermediate calculations will be carried out below the dividend, and you need to take care in advance about the availability of space on the page. In this case, you should be guided by the rule: the greater the difference in the number of characters in the entries of the dividend and divisor, the more space will be required. For example, when dividing by a column the natural number 614,808 by 51,234 (614,808 is a six-digit number, 51,234 is a five-digit number, the difference in the number of characters in the records is 6−5 = 1), intermediate calculations will require less space than when dividing the numbers 8 058 and 4 (here the difference in the number of characters is 4−1=3). To confirm our words, we present complete records of division by a column of these natural numbers:

Now you can proceed directly to the process of dividing natural numbers with a column.

Column division of a natural number by a single-digit natural number, column division algorithm

It is clear that dividing one single-digit natural number by another is quite simple, and there is no reason to divide these numbers into a column. However, it will be helpful to practice your initial long division skills with these simple examples.

Example.

Let us need to divide with a column of 8 by 2.

Solution.

Of course, we can perform division using the multiplication table, and immediately write down the answer 8:2=4.

But we are interested in how to divide these numbers with a column.

First, we write down the dividend 8 and the divisor 2 as required by the method:

Now we begin to find out how many times the divisor is contained in the dividend. To do this, we sequentially multiply the divisor by the numbers 0, 1, 2, 3, ... until the result is a number equal to the dividend (or a number greater than the dividend, if there is a division with a remainder). If we get a number equal to the dividend, then we immediately write it under the dividend, and in the place of the quotient we write the number by which we multiplied the divisor. If we get a number greater than the dividend, then under the divisor we write the number calculated at the penultimate step, and in place of the incomplete quotient we write the number by which the divisor was multiplied at the penultimate step.

Let's go: 2·0=0 ; 2 1=2 ; 2·2=4 ; 2·3=6 ; 2·4=8. We have received a number equal to the dividend, so we write it under the dividend, and in place of the quotient we write the number 4. In this case, the record will take the following form:

The final stage of dividing single-digit natural numbers with a column remains. Under the number written under the dividend, you need to draw a horizontal line, and subtract the numbers above this line in the same way as is done when subtracting natural numbers in a column. The resulting number after subtraction will be the remainder of the division. If it is equal to zero, then the original numbers are divided without a remainder.

In our example we get

Now we have before us a completed recording of the column division of the number 8 by 2. We see that the quotient of 8:2 is 4 (and the remainder is 0).

Answer:

8:2=4 .

Now let's look at how a column divides single-digit natural numbers with a remainder.

Example.

Divide with a column 7 by 3.

Solution.

At the initial stage, the entry looks like this:

We begin to find out how many times the dividend contains the divisor. We will multiply 3 by 0, 1, 2, 3, etc. until we get a number equal to or greater than the dividend 7. We get 3·0=0<7 ; 3·1=3<7 ; 3·2=6<7 ; 3·3=9>7 (if necessary, refer to the article comparing natural numbers). Under the dividend we write the number 6 (it was obtained at the penultimate step), and in place of the incomplete quotient we write the number 2 (the multiplication was carried out by it at the penultimate step).

It remains to carry out the subtraction, and the division by a column of single-digit natural numbers 7 and 3 will be completed.

Thus, the partial quotient is 2 and the remainder is 1.

Answer:

7:3=2 (rest. 1) .

Now you can move on to dividing multi-digit natural numbers by columns into single-digit natural numbers.

Now we'll figure it out long division algorithm. At each stage, we will present the results obtained by dividing the multi-digit natural number 140,288 by the single-digit natural number 4. This example was not chosen by chance, since when solving it we will encounter all possible nuances and will be able to analyze them in detail.

First we look at the first digit on the left in the dividend notation. If the number defined by this figure is greater than the divisor, then in the next paragraph we have to work with this number. If this number is less than the divisor, then we need to add to the consideration the next digit on the left in the notation of the dividend, and continue to work with the number determined by the two digits under consideration. For convenience, we highlight in our notation the number with which we will work.

The first digit from the left in the notation of the dividend 140288 is the digit 1. The number 1 is less than the divisor 4, so we also look at the next digit on the left in the notation of the dividend. At the same time, we see the number 14, with which we have to work further. We highlight this number in the notation of the dividend.

The following steps from the second to the fourth are repeated cyclically until the division of natural numbers by a column is completed.

Now we need to determine how many times the divisor is contained in the number we are working with (for convenience, let's denote this number as x). To do this, we sequentially multiply the divisor by 0, 1, 2, 3, ... until we get the number x or a number greater than x. When the number x is obtained, we write it under the highlighted number according to the recording rules used when subtracting natural numbers in a column. The number by which the multiplication was carried out is written in place of the quotient during the first pass of the algorithm (in subsequent passes of 2-4 points of the algorithm, this number is written to the right of the numbers already there). When we get a number that is greater than the number x, then under the highlighted number we write the number obtained at the penultimate step, and in place of the quotient (or to the right of the numbers already there) we write the number by which the multiplication was carried out at the penultimate step. (We carried out similar actions in the two examples discussed above).

Multiply the divisor 4 by the numbers 0, 1, 2, ... until we get a number that is equal to 14 or greater than 14. We have 4·0=0<14 , 4·1=4<14 , 4·2=8<14 , 4·3=12<14 , 4·4=16>14. Since at the last step we received the number 16, which is greater than 14, then under the highlighted number we write the number 12, which was obtained at the penultimate step, and in place of the quotient we write the number 3, since in the penultimate point the multiplication was carried out precisely by it.


At this stage, from the selected number, subtract the number located under it using a column. The result of the subtraction is written under the horizontal line. However, if the result of the subtraction is zero, then it does not need to be written down (unless the subtraction at that point is the very last action that completely completes the process of long division). Here, for your own control, it would not be amiss to compare the result of the subtraction with the divisor and make sure that it is less than the divisor. Otherwise, a mistake was made somewhere.

We need to subtract the number 12 from the number 14 with a column (for the correctness of the recording, we must remember to put a minus sign to the left of the numbers being subtracted). After completing this action, the number 2 appeared under the horizontal line. Now we check our calculations by comparing the resulting number with the divisor. Since the number 2 is less than the divisor 4, you can safely move on to the next point.


Now, under the horizontal line to the right of the numbers located there (or to the right of the place where we did not write down the zero), we write down the number located in the same column in the notation of the dividend. If there are no numbers in the record of the dividend in this column, then the division by column ends there. After this, we select the number formed under the horizontal line, accept it as a working number, and repeat points 2 to 4 of the algorithm with it.

Under the horizontal line to the right of the number 2 already there, we write down the number 0, since it is the number 0 that is in the record of the dividend 140,288 in this column. Thus, the number 20 is formed under the horizontal line.


We select this number 20, take it as a working number, and repeat with it the actions of the second, third and fourth points of the algorithm.

Multiply the divisor 4 by 0, 1, 2, ... until we get the number 20 or a number that is greater than 20. We have 4·0=0<20 , 4·1=4<20 , 4·2=8<20 , 4·3=12<20 , 4·4=16<20 , 4·5=20 . Так как мы получили число, равное числу 20 , то записываем его под отмеченным числом, а на месте частного, справа от уже имеющегося там числа 3 записываем число 5 (на него производилось умножение).


We carry out the subtraction in a column. Since we are subtracting equal natural numbers, then by virtue of the property of subtracting equal natural numbers, the result is zero. We do not write down the zero (since this is not the final stage of division with a column), but we remember the place where we could write it (for convenience, we will mark this place with a black rectangle).


Under the horizontal line to the right of the remembered place we write down the number 2, since it is precisely it that is in the record of the dividend 140,288 in this column. Thus, under the horizontal line we have the number 2.


We take the number 2 as the working number, mark it, and we will once again have to perform the actions of 2-4 points of the algorithm.

We multiply the divisor by 0, 1, 2, and so on, and compare the resulting numbers with the marked number 2. We have 4·0=0<2 , 4·1=4>2. Therefore, under the marked number we write the number 0 (it was obtained at the penultimate step), and in the place of the quotient to the right of the number already there we write the number 0 (we multiplied by 0 at the penultimate step).


We perform the subtraction in a column, we get the number 2 under the horizontal line. We check ourselves by comparing the resulting number with the divisor 4. Since 2<4 , то можно спокойно двигаться дальше.


Under the horizontal line to the right of the number 2, add the number 8 (since it is in this column in the entry for the dividend 140 288). Thus, the number 28 appears under the horizontal line.


We take this number as a working number, mark it, and repeat steps 2-4.

There shouldn't be any problems here if you have been careful up to now. Having completed all the necessary steps, the following result is obtained.

All that remains is to carry out the steps from points 2, 3, 4 one last time (we leave this to you), after which you will get a complete picture of dividing the natural numbers 140,288 and 4 into a column:

Please note that the number 0 is written in the very bottom line. If this was not the last step of division by a column (that is, if in the record of the dividend there were numbers left in the columns on the right), then we would not write this zero.

Thus, looking at the completed record of dividing the multi-digit natural number 140,288 by the single-digit natural number 4, we see that the quotient is the number 35,072 (and the remainder of the division is zero, it is in the very bottom line).

Of course, when dividing natural numbers by a column, you will not describe all your actions in such detail. Your solutions will look something like the following examples.

Example.

Perform long division if the dividend is 7 136 and the divisor is a single-digit natural number 9.

Solution.

At the first step of the algorithm for dividing natural numbers by columns, we get a record of the form

After performing the actions from the second, third and fourth points of the algorithm, the column division record will take the form

Repeating the cycle, we will have

One more pass will give us a complete picture of the column division of the natural numbers 7,136 and 9

Thus, the partial quotient is 792, and the remainder is 8.

Answer:

7 136:9=792 (rest. 8) .

And this example demonstrates what long division should look like.

Example.

Divide the natural number 7,042,035 by the single-digit natural number 7.

Solution.

The most convenient way to do division is by column.

Answer:

7 042 035:7=1 006 005 .

Column division of multi-digit natural numbers

Let us hasten to please you: if you have thoroughly mastered the column division algorithm from the previous paragraph of this article, then you almost already know how to perform column division of multi-digit natural numbers. This is true, since stages 2 to 4 of the algorithm remain unchanged, and only minor changes appear in the first point.

At the first stage of dividing multi-digit natural numbers into a column, you need to look not at the first digit on the left in the notation of the dividend, but at the number of them equal to the number of digits contained in the notation of the divisor. If the number defined by these numbers is greater than the divisor, then in the next paragraph we have to work with this number. If this number is less than the divisor, then we need to add to the consideration the next digit on the left in the notation of the dividend. After this, the actions specified in paragraphs 2, 3 and 4 of the algorithm are performed until the final result is obtained.

All that remains is to see the application of the column division algorithm for multi-valued natural numbers in practice when solving examples.

Example.

Let's perform column division of multi-digit natural numbers 5,562 and 206.

Solution.

Since the divisor 206 contains 3 digits, we look at the first 3 digits on the left in the dividend 5,562. These numbers correspond to the number 556. Since 556 is greater than the divisor 206, we take the number 556 as a working number, select it, and move on to the next stage of the algorithm.

Now we multiply the divisor 206 by the numbers 0, 1, 2, 3, ... until we get a number that is either equal to 556 or greater than 556. We have (if multiplication is difficult, then it is better to multiply natural numbers in a column): 206 0 = 0<556 , 206·1=206<556 , 206·2=412<556 , 206·3=618>556. Since we received a number that is greater than the number 556, then under the highlighted number we write the number 412 (it was obtained at the penultimate step), and in place of the quotient we write the number 2 (since we multiplied by it at the penultimate step). The column division entry takes the following form:

We perform column subtraction. We get the difference 144, this number is less than the divisor, so you can safely continue performing the required actions.

Under the horizontal line to the right of the number there we write the number 2, since it is in the record of the dividend 5562 in this column:

Now we work with the number 1,442, select it, and go through steps two through four again.

Multiply the divisor 206 by 0, 1, 2, 3, ... until you get the number 1442 or a number that is greater than 1442. Let's go: 206·0=0<1 442 , 206·1=206<1 442 , 206·2=412<1 332 , 206·3=618<1 442 , 206·4=824<1 442 , 206·5=1 030<1 442 , 206·6=1 236<1 442 , 206·7=1 442 . Таким образом, под отмеченным числом записываем 1 442 , а на месте частного правее уже имеющегося там числа записываем 7 :

We carry out the subtraction in a column, we get zero, but we don’t write it down right away, we just remember its position, because we don’t know whether the division ends here, or whether we’ll have to repeat the steps of the algorithm again:

Now we see that we cannot write any number under the horizontal line to the right of the remembered position, since there are no digits in the record of the dividend in this column. Therefore, this completes the division by column, and we complete the entry:

• Mathematics. Any textbooks for 1st, 2nd, 3rd, 4th grades of general education institutions.
• Mathematics. Any textbooks for 5th grade of general education institutions.

The easiest way to divide multi-digit numbers is with a column. Column division is also called corner division.

Before we begin performing division by a column, we will consider in detail the very form of recording division by a column. First, we write down the dividend and put a vertical line to the right of it:

Behind the vertical line, opposite the dividend, write the divisor and draw a horizontal line under it:

Under the horizontal line, the resulting quotient will be written step by step:

Intermediate calculations will be written under the dividend:

The full form of writing division by column is as follows:

How to divide by column

Let's say we need to divide 780 by 12, write the action in a column and proceed to division:

Column division is performed in stages. The first thing we need to do is determine the incomplete dividend. We look at the first digit of the dividend:

this number is 7, since it is less than the divisor, we cannot start division from it, which means we need to take another digit from the dividend, the number 78 is greater than the divisor, so we start division from it:

In our case the number 78 will be incomplete divisible, it is called incomplete because it is only a part of the divisible.

Having determined the incomplete dividend, we can find out how many digits will be in the quotient, for this we need to calculate how many digits are left in the dividend after the incomplete dividend, in our case there is only one digit - 0, this means that the quotient will consist of 2 digits.

Having found out the number of digits that should be in the quotient, you can put dots in its place. If, when completing the division, the number of digits turns out to be more or less than the indicated points, then an error was made somewhere:

Let's start dividing. We need to determine how many times 12 is contained in the number 78. To do this, we sequentially multiply the divisor by the natural numbers 1, 2, 3, ... until we get a number as close as possible to the incomplete dividend or equal to it, but not exceeding it. Thus, we get the number 6, write it under the divisor, and from 78 (according to the rules of column subtraction) we subtract 72 (12 6 = 72). After we subtract 72 from 78, the remainder is 6:

Please note that the remainder of the division shows us whether we have chosen the number correctly. If the remainder is equal to or greater than the divisor, then we did not choose the number correctly and we need to take a larger number.

To the resulting remainder - 6, add the next digit of the dividend - 0. As a result, we get an incomplete dividend - 60. Determine how many times 12 is contained in the number 60. We get the number 5, write it in the quotient after the number 6, and subtract 60 from 60 ( 12 5 = 60). The remainder is zero:

Since there are no more digits left in the dividend, it means 780 is divided by 12 completely. As a result of performing long division, we found the quotient - it is written under the divisor:

Let's consider an example when the quotient turns out to be zeros. Let's say we need to divide 9027 by 9.

We determine the incomplete dividend - this is the number 9. We write 1 into the quotient and subtract 9 from 9. The remainder is zero. Usually, if in intermediate calculations the remainder is zero, it is not written down:

We take down the next digit of the dividend - 0. We remember that when dividing zero by any number there will be zero. We write zero into the quotient (0: 9 = 0) and subtract 0 from 0 in intermediate calculations. Usually, in order not to clutter up intermediate calculations, calculations with zero are not written:

We take down the next digit of the dividend - 2. In intermediate calculations it turned out that the incomplete dividend (2) is less than the divisor (9). In this case, write zero to the quotient and remove the next digit of the dividend:

We determine how many times 9 is contained in the number 27. We get the number 3, write it as a quotient, and subtract 27 from 27. The remainder is zero:

Since there are no more digits left in the dividend, it means that the number 9027 is divided by 9 completely:

Let's consider an example when the dividend ends in zeros. Let's say we need to divide 3000 by 6.

We determine the incomplete dividend - this is the number 30. We write 5 into the quotient and subtract 30 from 30. The remainder is zero. As already mentioned, it is not necessary to write zero in the remainder in intermediate calculations:

We take down the next digit of the dividend - 0. Since dividing zero by any number will result in zero, we write zero in the quotient and subtract 0 from 0 in intermediate calculations:

We take down the next digit of the dividend - 0. We write another zero into the quotient and subtract 0 from 0 in intermediate calculations. Since in intermediate calculations the calculation with zero is usually not written down, the entry can be shortened, leaving only the remainder - 0. Zero in the remainder in at the very end of the calculation is usually written to show that the division is complete:

Since there are no more digits left in the dividend, it means 3000 is divided by 6 completely:

Column division with remainder

Let's say we need to divide 1340 by 23.

We determine the incomplete dividend - this is the number 134. We write 5 into the quotient and subtract 115 from 134. The remainder is 19:

We take down the next digit of the dividend - 0. We determine how many times 23 is contained in the number 190. We get the number 8, write it into the quotient, and subtract 184 from 190. We get the remainder 6:

Since there are no more digits left in the dividend, the division is over. The result is an incomplete quotient of 58 and a remainder of 6:

1340: 23 = 58 (remainder 6)

It remains to consider an example of division with a remainder, when the dividend is less than the divisor. Let us need to divide 3 by 10. We see that 10 is never contained in the number 3, so we write 0 as a quotient and subtract 0 from 3 (10 · 0 = 0). Draw a horizontal line and write down the remainder - 3:

3: 10 = 0 (remainder 3)

Long division calculator

This calculator will help you perform long division. Simply enter the dividend and divisor and click the Calculate button.

Children in grades 2-3 are learning a new mathematical operation - division. It is not easy for a student to understand the essence of this mathematical operation, so he needs the help of his parents. Parents need to understand exactly how to present new information to their child. TOP 10 examples will tell parents how to teach children how to divide numbers in a column.

Learning long division in the form of a game

Children get tired at school, they get tired of textbooks. Therefore, parents need to give up textbooks. Present information in the form of a fun game.

You can set tasks this way:

1 Organize a place for your child to learn through play. Place his toys in a circle, and give the child pears or candy. Have the student divide 4 candies between 2 or 3 dolls. To achieve understanding on the part of the child, gradually increase the number of candies to 8 and 10. Even if the baby takes a long time to act, do not put pressure or yell at him. You will need patience. If your child does something wrong, correct him calmly. Then, after he completes the first action of dividing the candies between the participants in the game, he will ask him to calculate how many candies went to each toy. Now the conclusion. If there were 8 candies and 4 toys, then each got 2 candies. Let your child understand that sharing means distributing an equal amount of candy to all toys.

2 You can teach math using numbers. Let the student understand that numbers can be classified as pears or candy. Say that the number of pears to be divided is the dividend. And the number of toys that contain candy is the divisor.

3 Give your child 6 pears. Give him a task: to divide the number of pears between grandfather, dog and dad. Then ask him to divide 6 pears between grandpa and dad. Explain to your child the reason why the division result was different.

4 Teach your student about division with a remainder. Give your child 5 candies and ask him to distribute them equally between the cat and dad. The child will have 1 candy left. Tell your child why it happened this way. This mathematical operation should be considered separately, as it can cause difficulties.

Playful learning can help your child quickly understand the whole process of dividing numbers. He will be able to learn that the largest number is divisible by the smallest or vice versa. That is, the largest number is candy, and the smallest number is the participants. In column 1 the number will be the number of candies, and 2 will be the number of participants.

Do not overload your child with new knowledge. You need to learn gradually. You need to move on to new material when the previous material is consolidated.

Learning long division using the multiplication table

Students up to 5th grade will be able to understand division more quickly, provided they have a good understanding of multiplication.

Parents need to explain that division is similar to the multiplication table. Only the actions are opposite. For clarity, we need to give an example:

• Tell the student to freely multiply the values ​​of 6 and 5. The answer is 30.
• Tell the student that the number 30 is the result of a mathematical operation with two numbers: 6 and 5. Namely, the result of multiplication.
• Divide 30 by 6. The result of the mathematical operation is 5. The student will be able to see that division is the same as multiplication, but in reverse.

You can use the multiplication table to illustrate division if the child has mastered it well.

Learning long division in a notebook

Learning should begin when the student understands the material about division in practice, using games and multiplication tables.

You need to start dividing in this way, using simple examples. So, divide 105 by 5.

The mathematical operation needs to be explained in detail:

• Write an example in your notebook: 105 divided by 5.
• Write this down as you would for long division.
• Explain that 105 is the dividend and 5 is the divisor.
• With a student, identify 1 number that can be divided. The value of the dividend is 1, this figure is not divisible by 5. But the second number is 0. The result is 10, this value can be divided in this example. The number 5 is included in the number 10 twice.
• In the division column, under the number 5, write the number 2.
• Ask your child to multiply the number 5 by 2. The result of the multiplication is 10. This value must be written under the number 10. Next, you need to write the subtraction sign in the column. From 10 you need to subtract 10. You get 0.
• Write down in the column the number resulting from the subtraction - 0. 105 has a number left that was not involved in the division - 5. This number needs to be written down.
• The result is 5. This value must be divided by 5. The result is the number 1. This number must be written under 5. The result of the division is 21.

Parents need to explain that this division has no remainder.

You can start division with numbers 6,8,9, then go to 22, 44, 66 , and then to 232, 342, 345 , and so on.

Learning division with remainder

Once the child has mastered the material about division, you can make the task more difficult. Division with a remainder is the next step in learning. You need to explain using available examples:

• Invite your child to divide 35 by 8. Write the problem in the column.
• To make it as clear as possible for your child, you can show him the multiplication table. The table clearly shows that the number 35 includes the number 8 4 times.
• Write down the number 32 under the number 35.
• The child needs to subtract 32 from 35. The result is 3. The number 3 is the remainder.

Simple examples for a child

We can continue with the same example:

• When dividing 35 by 8, the remainder is 3. You need to add 0 to the remainder. In this case, after the number 4 in the column you need to put a comma. Now the result will be fractional.
• When dividing 30 by 8, the result is 3. This number must be written after the decimal point.
• Now you need to write 24 under the value 30 (the result of multiplying 8 by 3). The result will be 6. You also need to add a zero to the number 6. It will turn out to be 60.
• The number 60 contains the number 8 included 7 times. That is, it turns out to be 56.
• When subtracting 60 from 56, the result is 4. This number also needs to be signed 0. The result is 40. In the multiplication table, a child can see that 40 is the result of multiplying 8 by 5. That is, the number 40 includes the number 8 5 times. There is no remainder. The answer looks like this - 4.375.

This example may seem difficult to a child. Therefore, you need to divide values ​​that will have a remainder many times.

Teaching division through games

Parents can use division games to teach their students. You can give your child coloring books in which you need to determine the color of a pencil by dividing. You need to choose coloring pages with easy examples so that the child can solve the examples in his head.

The picture will be divided into parts containing the results of the division. And the colors to use will be examples. For example, the color red is labeled with an example: 15 divided by 3. You get 5. You need to find the part of the picture under this number and color it. Math coloring pages captivate children. Therefore, parents should try this method of teaching.

Learning to divide by column the smallest number by the largest

Division by this method assumes that the quotient will start at 0 and will be followed by a comma.

In order for the student to correctly assimilate the information received, he needs to give an example of such a plan.

Column? How can you independently practice the skill of long division at home if your child did not learn something at school? Dividing by columns is taught in grades 2-3; for parents, of course, this is a passed stage, but if you wish, you can remember the correct notation and explain in an understandable way to your student what he will need in life.

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What should a 2nd-3rd grade child know to learn to do long division?

How to correctly explain division to a 2-3 grade child so that he doesn’t have problems in the future? First, let's check if there are any gaps in knowledge. Make sure that:

• the child can freely perform addition and subtraction operations;
• knows the digits of numbers;
• knows by heart.

How to explain to a child the meaning of the action “division”?

• Everything needs to be explained to the child using a clear example.

Ask to share something among family members or friends. For example, candy, pieces of cake, etc. It is important that the child understands the essence - you need to divide equally, i.e. without a trace. Practice with different examples.

Let's say 2 groups of athletes must take seats on the bus. We know how many athletes are in each group and how many seats there are on the bus. You need to find out how many tickets one and the other group need to buy. Or 24 notebooks should be distributed to 12 students, as many as each gets.

• When the child understands the essence of the principle of division, show the mathematical notation of this operation and name the components.
• Explain that Division is the opposite operation of multiplication, multiplication inside out.

It is convenient to show the relationship between division and multiplication using a table as an example.

For example, 3 times 4 equals 12.
3 is the first multiplier;
4 - second factor;
12 is the product (the result of multiplication).

If 12 (the product) is divided by 3 (the first factor), we get 4 (the second factor).

Components when divided are called differently:

12 - dividend;
3 - divider;
4 - quotient (result of division).

How to explain to a child the division of a two-digit number by a single-digit number not in a column?

For us adults, it’s easier to write “in the corner” the old fashioned way – and that’s the end of it. BUT! Children have not yet completed long division, what should they do? How to teach a child to divide a two-digit number by a single-digit number without using column notation?

Let's take 72:3 as an example.

It's simple! We break down 72 into numbers that can be easily divided verbally by 3:
72=30+30+12.

Everything immediately became clear: we can divide 30 by 3, and a child can easily divide 12 by 3.
All that remains is to add up the results, i.e. 72:3=10 (obtained when 30 was divided by 3) + 10 (30 divided by 3) + 4 (12 divided by 3).

72:3=24
We did not use long division, but the child understood the reasoning and completed the calculations without difficulty.

After simple examples, you can move on to studying long division and teach your child to correctly write examples in a “corner”. To begin with, use only examples of division without a remainder.

How to explain long division to a child: solution algorithm

Large numbers are difficult to divide in your head; it is easier to use column division notation. To teach your child to perform calculations correctly, follow the algorithm:

• Determine where the dividend and divisor are in the example. Ask your child to name the numbers (what we will divide by what).

213:3
213 - dividend
3 - divider

• Write down the dividend - "corner" - divisor.

• Determine which part of the dividend we can use to divide by a given number.

We reason like this: 2 is not divisible by 3, which means we take 21.

• Determine how many times the divisor “fits” in the selected part.

21 divided by 3 - take 7.

• Multiply the divisor by the selected number, write the result under the “corner”.

7 multiplied by 3 - we get 21. Write it down.

• Find the difference (remainder).

At this stage of reasoning, teach your child to check himself. It is important that he understands that the result of a subtraction must ALWAYS be less than the divisor. If it doesn't work out, you need to increase the selected number and perform the action again.

• Repeat the steps until the remainder is 0.

How to reason correctly to teach a 2-3 grade child to divide by column

How to explain division to a child 204:12=?
1. Write it down in a column.
204 is the dividend, 12 is the divisor.

2. 2 is not divisible by 12, so we take 20.
3. To divide 20 by 12, take 1. Write 1 under the “corner”.
4. 1 multiplied by 12 gets 12. We write it under 20.
5. 20 minus 12 gets 8.
Let's check ourselves. Is 8 less than 12 (divisor)? Ok, that's right, let's move on.

6. Next to 8 we write 4. 84 divided by 12. How much should we multiply 12 to get 84?
It’s hard to say right away, we’ll try to use the selection method.
Let's take 8, for example, but don't write them down yet. We count verbally: 8 multiplied by 12 equals 96. And we have 84! Doesn't fit.
Let's try smaller ones... For example, let's take 6 each. We check ourselves verbally: 6 multiplied by 12 equals 72. 84-72 = 12. We got the same number as our divisor, but it should be either zero or less than 12. So the optimal number is 7!

7. We write 7 under the “corner” and perform the calculations. 7 multiplied by 12 gives 84.
8. We write the result in a column: 84 minus 84 equals zero. Hooray! We decided correctly!

So, you have taught your child to divide by column, now all that remains is to practice this skill and bring it to automatism.

Why is it difficult for children to learn long division?

Remember that problems with mathematics arise from the inability to quickly do simple arithmetic operations. In elementary school, you need to practice addition and subtraction and make it automatic, and learn the multiplication table from cover to cover. All! The rest is a matter of technique, and it is developed with practice.

Be patient, do not be lazy, once again explain to the child what he did not learn in the lesson, tediously but meticulously understand the reasoning algorithm and talk through each intermediate operation before voicing a ready answer. Give additional examples to practice skills, play math games - this will bear fruit and you will see the results and rejoice at your child’s success very soon. Be sure to show where and how you can apply the acquired knowledge in everyday life.

Dear readers! Tell us how you teach your children to do long division, what difficulties you have encountered and how you have overcome them.

How to do long division is one of the basic skills needed to work with two- and three-digit numbers. Knowing the sequence of all stages of division, you can divide any number. There will be no problems when working not only with an integer number, but also with a number presented as a decimal fraction.

This useful mathematical skill is necessary not only for successfully mastering the school curriculum in mathematics and a number of other subjects. The ability to share will certainly help everyone in everyday life.

Part one. Division

So, the dividend, that is, the number that needs to be divided, must be written on the left. The number being divided is called the divisor and is written on the right.

A line is drawn under the divisor, under which the quotient (solution) is written.

Under the dividend, you must leave the space required for calculations.

The problem itself looks like this: a bag containing six mushrooms weighs 250 grams. You need to find out how much one mushroom weighs. To do this, 250 is divided by 6. The first of these two numbers is written on the left, and the second on the right.

Now we need to calculate how many integer times the first digit is divisible (counting from the left end) of the divisor.

To solve our problem, we need to find out how many times the number 2 is divisible by 6. Since this is impossible, the answer is 0, which is written under the divisor. In this case, zero is the first number of the quotient, but it is possible to refuse such an entry.

Now we need to find out how many times the first two digits of the dividend are divided by the divisor.

If in the previous action the answer was 0, you need to consider the first two digits of the dividend. In the problem under consideration, we need to calculate how many times 25 is divisible by 6.

If the divisor is a two- or more-digit number, you must divide the first three (four, five, etc.) digits of the dividend by it. Our goal: get an integer.

Next we begin working with integers. If you use a microcalculator to divide 25 by 6, the answer will be 4.167. This answer is not suitable for long division. In this case you just need to take 4.

The result obtained in the third stage is written directly under the corresponding digit of the divisor - under the line. This total will be the first digit of the desired quotient, that is, the answer.

The result must be written under the corresponding digit of the divisor. If you neglect this requirement, a mistake will be made, which will affect the final result: it will be incorrect.

In this case, 4 is written under 5, since 6 is divisible by 25, not 2.

Part two. Multiplication

This stage represents a transition to a new part of the work “how to count in a column.” Division in this case will be replaced by... multiplication.

The divisor is multiplied by the number that was written below it. This means that we are talking about the first digit of the desired quotient.

The result of this product is placed under the dividend.

In the example under consideration, 6 x 4 = 24. The number in the answer, that is, 24, is written under 25. Important: 2 must be under 2, and 4 must be under 5.

The result of the work is emphasized. In our case, we are talking about emphasizing the number 24.

Part three. Subtracting and omitting numbers

This is where the transition to subtracting and lowering numbers occurs.

The result is written under the line, which in turn is drawn under the number placed under the dividend.

We have to subtract 24 from 25. The result we get is: 1.

The third digit of the dividend is omitted, that is, it is written next to the result of the subtraction.

In our case, 1 cannot be divided by 6. Because of this, the third digit of the dividend is omitted (the third digit of the number 250 is 0). It is placed next to 1. We get the number 10, which can be divided by 6.

Now you need to repeat the process with a new number.

To do this, the resulting number is divided by our divisor, and the result obtained is placed under the divisor, which will be the second digit of the quotient, that is, our answer.

In the example being solved, we divide 10 by 6, which gives a total of 1. One is written into the quotient - next to 4. After this, 6 is multiplied by 1 and the result is subtracted from 10. We should get 4 (remainder).

If the dividend is a two-, three-, four-, or more-digit number, the above process is repeated until all the digits of the dividend are omitted. An example to illustrate: if you know that the weight of mushrooms is 2,506 g, you need to omit the number 6, that is, write it next to 4.

Part four. Writing a quotient with a remainder or as a decimal fraction

Now we move on to writing the quotient with a remainder or in the form of a decimal fraction.

Our remainder was equal to 4, which is due to the fact that this number - 4 - is not divisible by 6 and we have no numbers left that can be omitted.

The answer will look like this: 41 (rest. 4).

Calculations at this stage can be completed if the problem requires finding something that can be expressed exclusively in integers. We can talk about the number of cars required to transport a certain number of people.

If there is a need for an answer in the form of a decimal fraction, you can proceed to the next steps of the “how to divide into a column” algorithm.

If you do not want to write the answer with a remainder, you can find the answer in the form of a decimal fraction. When obtaining a remainder that cannot be divided by a divisor, you must add a decimal sign (to the quotient).

In our case, the number 250 can be written as a decimal fraction: 250.000.

Now that there are numbers (only zeros) that can be omitted, we can continue the calculations. We omit the zero and count how many times the resulting number can be divided by the divisor.

In our example, after the quotient 41 (which we place directly below the divisor), we write a decimal point and add 0 to the remainder (4). Then we divide the resulting number, that is, 40, by the divisor (which is 6). We get 6 again, which we write as the quotient after the decimal place. It looks like 41.6. After this, 6 is multiplied by 6, then the result of the multiplication is subtracted from 40. We should get 4 again.

In a number of situations, when searching for an answer in the form of a decimal fraction, you may encounter repeating numbers. To do this, you need to interrupt the calculations and round the answer you have already received - down or up.

In particular, in the example under consideration, we need to stop endlessly obtaining the number 4. We just need to interrupt the calculations and round the quotient. Because 6 is greater than 5, rounding is done upward, resulting in a fractional answer of 41.67.