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

Before we begin to perform division by a column, we will consider in detail the very form of recording division by a column. First, 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 complete 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 results in 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.

Column division is an integral part educational material junior school student. Further success in mathematics will depend on how correctly he learns to perform this action.

How to properly prepare a child to perceive new material?

Column division is a complex process that requires certain knowledge from the child. To perform division, you need to know and be able to quickly subtract, add, and multiply. Knowledge of number digits is also important.

Each of these actions should be brought to automaticity. The child should not have to think for a long time, and also be able to subtract and add not only numbers from the first ten, but within a hundred in a few seconds.

It is important to form correct concept division as a mathematical operation. Even when studying multiplication and division tables, the child must clearly understand that the dividend is a number that will be divided into equal parts, the divisor indicates how many parts the number should be divided into, and the quotient is the answer itself.

How to explain the algorithm of a mathematical operation step by step?

Each mathematical operation requires strict adherence to a specific algorithm. Examples of long division should be performed in this order:

1. Write the example in a corner, and the places of the dividend and divisor must be strictly observed. To help the child not get confused in the first stages, we can say that we write a larger number on the left and a smaller number on the right.
2. Select a part for the first division. It must be divisible by the dividend with a remainder.
3. Using the multiplication table, we determine how many times the divisor can fit in the selected part. It is important to indicate to the child that the answer should not exceed 9.
4. Multiply the resulting number by the divisor and write it on the left side of the corner.
5. Next, you need to find the difference between the part of the dividend and the resulting product.
6. The resulting number is written below the line and the next digit number is taken down. Such actions are performed until the remainder is 0.

A clear example for students and parents

Column division can be clearly explained using this example.

1. Write down 2 numbers in a column: the dividend is 536 and the divisor is 4.
2. The first part for division must be divisible by 4 and the quotient must be less than 9. The number 5 is suitable for this.
3. 4 fits into 5 only once, so we write 1 in the answer, and 4 under 5.
4. Next, subtraction is performed: 4 is subtracted from 5 and 1 is written under the line.
5. The next digit number is added to one - 3. In thirteen (13) - 4 fits 3 times. 4x3 = 12. Twelve is written under the 13th, and 3 is written as the quotient, as the next digit number.
6. 12 is subtracted from 13, the answer is 1. The next digit number is taken away again - 6.
7. 16 is again divided by 4. The answer is written as 4, and in the division column - 16, and the difference is drawn as 0.

By solving long division examples with your child several times, you can achieve success in quickly completing problems in middle school.

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 to their child new information. 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 operations 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.

Training in game form can help your child quickly understand the whole process of dividing numbers. He will be able to understand that greatest number divided 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 a 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 did not participate in the division - 5. This number must 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. So parents should try this method training.

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.

Instructions

Before teaching how to divide two-digit numbers, you need to explain to your child that a number is the sum of tens and units. This will save him from future quite a common mistake that many children make. They begin to divide the first and second digits of the dividend and divisor by each other.

First, work from numbers to single digits. This technique is best practiced using knowledge of the multiplication table. The more such practice there is, the better. The skills of such division should be brought to automaticity, then it will be easier for the child to move on to the more complex topic of the two-digit divisor, which, like the dividend, is the sum of tens and units.

The most common method of dividing two-digit numbers is the brute method, which involves successively dividing numbers from 2 to 9 so that the resulting product equals the dividend. Example: divide 87 by 29. Reason as follows:

29 times 2 equals 54 – not enough;
29 x 3 = 87 – correct.

Draw the student's attention to the second digits (units) of the dividend and divisor, which are convenient to focus on when using the multiplication table. For example, in the above example, the second digit of the divisor is 9. Think about how much you need to multiply the number 9 so that the number of units of the product equals 7? Answer in in this case only one – for 3. This makes the task much easier two-digit division. Test your guess by multiplying the entire number 29.

If the task is completed in writing, then it is advisable to use the column division method. This approach is similar to the previous one except that the student does not need to keep the numbers in his head and do mental calculations. It is better to arm yourself with a pencil or a rough piece of paper for written work.

Sources:

• multiplying two-digit numbers by two-digit tables

The topic of dividing numbers is one of the most important in the 5th grade math program. Without mastering this knowledge, further study of mathematics is impossible. Divide numbers happen in life every day. And you shouldn’t always rely on a calculator. To divide two numbers, you need to remember a certain sequence of actions.

You will need

• A sheet of paper in a square,
• pen or pencil

Instructions

Write down the dividend on one line. Separate them with a vertical line two lines high. Draw a horizontal line under the divisor and dividend perpendicular to the previous line. The quotient will be written to the right under this line. Below and to the left of the dividend, under the horizontal line, write down a zero.

Move the one leftmost, but not yet transferred, digit of the dividend down below the last horizontal line. Mark the transferred digit of the dividend with a dot.

Compare the number under the last horizontal line with the divisor. If the number is less than the divisor then continue from step 4, otherwise go to step 5.

Column division(you can also find the name division corner) - standard procedure Varithmetic, designed to divide simple or complex multi-digit numbers by breakingdividing by a series of more simple steps. As with all division problems, one number, calleddivisible, is divided into another, calleddivider, producing a result calledprivate.

The column can be used to divide natural numbers without a remainder, as well as to divide natural numbers with the remainder.

Rules for writing when dividing by a column.

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

First, the dividend and the divisor are written in one line from left to right, then between the writtennumbers represent a symbol of the form.

For example, if the dividend is 6105 and the divisor is 55, then their correct notation when dividing inthe column will be like this:

Look at the following diagram illustrating places to write dividend, divisor, quotient,remainder and intermediate calculations when dividing by a column:

From the above diagram it is clear that the required quotient (or incomplete quotient when divided with a remainder) will bewritten below the divisor under the horizontal bar. And intermediate calculations will be carried out belowdivisible, and you need to take care in advance about the availability of space on the page. In this case, one should be guidedrule: the greater the difference in the number of characters in the entries of the dividend and the divisor, the greaterspace will be required.

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

How to do long division is best explained with an example.Calculate:

512:8=?

First, let's write down the dividend and divisor in a column. It will look like this:

We will write their quotient (result) under the divisor. For us this is number 8.

1. Define an incomplete quotient. 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 workwith this number. If this number is less than the divisor, then we need to add the following to considerationon the left the figure in the notation of the dividend, and work further with the number determined by the two consideredin numbers. For convenience, we highlight in our notation the number with which we will work.

2. Take 5. The number 5 is less than 8, which means you need to take one more number from the dividend. 51 is greater than 8. So.this is an incomplete quotient. We put a dot in the quotient (under the corner of the divisor).

After 51 there is only one number 2. This means we add one more point to the result.

3. Now, remembering multiplication table by 8, find the product closest to 51 → 6 x 8 = 48→ write the number 6 into the quotient:

We write 48 under 51 (if we multiply 6 from the quotient by 8 from the divisor, we get 48).

Attention! When writing under an incomplete quotient, the rightmost digit of the incomplete quotient should be aboverightmost digit works.

4. Between 51 and 48 on the left we put “-” (minus). Subtract according to the rules of subtraction in column 48 and below the lineLet's write down the result.

However, if the result of the subtraction is zero, then it does not need to be written (unless the subtraction is inthis point is not the very last action that completely completes the division process column).

The remainder is 3. Let's compare the remainder with the divisor. 3 is less than 8.

Attention!If the remainder is greater than the divisor, then we made a mistake in the calculation and the product iscloser than the one we took.

5. Now, under the horizontal line to the right of the numbers located there (or to the right of the place where we do notbegan to write down zero) we write down the number located in the same column in the record of the dividend. If inThere are no numbers in the dividend entry in this column, then division by column ends here.

The number 32 is greater than 8. And again, using the multiplication table by 8, we find the nearest product → 8 x 4 = 32:

The remainder was zero. This means that the numbers are completely divided (without remainder). If after the lastsubtraction results in zero, and there are no more digits left, then this is the remainder. We add it to the quotient inparentheses (eg 64(2)).

Column division of multi-digit natural numbers.

Division by natural multi-digit number produced in the same way. At the same time, in the firstThe “intermediate” dividend includes so many high-order digits that it becomes larger than the divisor.

For example, 1976 divided by 26.

• The number 1 in the most significant digit is less than 26, so consider a number made up of two digits senior ranks - 19.
• The number 19 is also less than 26, so consider a number made up of the digits of the three highest digits - 197.
• The number 197 is greater than 26, divide 197 tens by 26: 197: 26 = 7 (15 tens left).
• Convert 15 tens to units, add 6 units from the units digit, we get 156.
• Divide 156 by 26 to get 6.

So 1976: 26 = 76.

If at some division step the “intermediate” dividend turns out to be less than the divisor, then in the quotient0 is written, and the number from of this category transferred to the next, lower rank.

Division with decimal fraction in quotient.

Decimals online. Translation decimals in ordinary and ordinary fractions to decimals.

If the natural number is not divisible by a single digit natural number, you can continuebitwise division and get a decimal fraction in the quotient.

For example, divide 64 by 5.

• Divide 6 tens by 5, we get 1 ten and 1 ten as a remainder.
• We convert the remaining ten into units, add 4 from the ones category, and get 14.
• We divide 14 units by 5, we get 2 units and a remainder of 4 units.
• We convert 4 units to tenths, we get 40 tenths.
• Divide 40 tenths by 5 to get 8 tenths.

So 64:5 = 12.8

Thus, if upon division natural number to a natural single-digit or multi-digit numberthe remainder is obtained, then you can put a comma in the quotient, convert the remainder into units of the following,smaller digit and continue dividing.