The basics of division in a column and in the mind, children study in elementary school: in the 3rd or 4th grade. But far from all third-graders delve into the material quickly and easily. At home, you need to practice a lot, solve training examples. But first, it is better to explain the division by a corner again, with a remainder, to identify gaps in children's knowledge.

How to become a super teacher without special training and help a child with this difficult topic, we will tell you in more detail.

How to learn to share a column

Division by a column with and without a remainder cannot be started without preparation. First, the child should be well able and aware of the following:

Practice all the indicated skills to automatism. Then proceed to divide small numbers using the mental multiplication table as an example. For example, a child learned how to multiply the number 6:

Feel free to offer examples such as:

After a couple of lessons, the student will perform such tasks easily. You can diversify your mental counting lessons with division games.

On a note! All initial mathematical skills are well automated with the help of online tests, where the child receives an instant result of his work.

Game tasks

Interesting mathematical division games help children to consolidate the skill, learn the laws of working with numbers, master mental counting.

• Puzzles for the development of attention. Write in your notebook 3-5 examples per division with answers. All but one must be solved incorrectly. You need to quickly find the example that contains the correct answer. Then correct the rest with a mental count.
• Selection of an example by result. Offer the child an answer without an example. Let's come up with a task. For example, the answer is 8. The child can come up with the following problem: 48:6.
• "Let's go to the store." Place toys with cards on the floor. Examples are written on the sheets: 6:2, 18:3, 42:7, 100:50. Toys are the "goods" in the fantasy store, the quotient after solving the example is their price. To find out the cost of the purchase, you need to solve the tasks, and then pay the result to the cashier. It is better to play in a small team - 2-3 people.
• "Silent". The child receives cards with numbers from 1 to 100. Ask questions with examples for division, the student must answer without words, showing the correct answer.
• Small independent work with a gift for diligence. Print out 5-10 sample cards. Specify the time for the solution, for example 5 minutes. Place an hourglass in front of your child. After completing the control correctly, encourage the student to go to the zoo, cinema, buy a book, sweets.
• "Looking for a tree." Draw a small garden with trees on cardboard. Give each plant a number, let there be 10 of them. Write 3 examples on a piece of paper for the student:

45:9 120:60 14:7

The student must calculate the result for each task, and then add all the numbers together. It will turn out like this:

The child must find the tree at number 9.

For the game, you can use colored buttons and put them on busy trees. Entertainment suitable for team competitions.

After oral work with the division of natural numbers, you can show the child the order in which examples are written in a column. If you do not have pedagogical experience, watch a video lesson on this topic, remember the theory yourself.

Now you can begin to explain the complex material to the student. There are several methods for home teaching division:

1. Mother teacher

Parents will have to become teachers for a while. Equip the board, buy chalk or markers. Recall the school material in advance. Explain the theory step by step and consolidate it in practice with the help of a large number of independent, cards, tests.

2. Watch an educational video with your child

For example, this:

Then you need to discuss the material with the baby, consolidate the skill in practice for several weeks.

3. Hire a tutor

Division is not the most difficult topic in the school curriculum. In elementary grades, you can easily do without paid lessons with a teacher. We leave this option as a last resort.

On a note! Be sure to contrast division with multiplication. Check the result of both actions by the opposite.

How to explain column division

First, it is worth explaining intelligibly what division is with a simple example. The essence of the mathematical operation is to decompose the number equally. In the 3rd grade, children learn well from available examples: they distribute pieces of cake to guests, they seat dolls in 2 cars.

When the baby learns the essence of the division, show his record on the sheet. Use already familiar tasks with prime numbers:

• First, write down the task in the usual way: 250:2=?
• Give each number a name: 250 is the dividend, 2 is the divisor, the result after the equal sign is the quotient.
• Then make an abbreviated entry in a column (corner):

• Argue together like this: first we find an incomplete quotient. This will be 2, since it is not less than the divisor, or rather, equal to it. One divisor is placed in this number, which means that we write the number 1 into the quotient and multiply it by 2. We enter the result obtained under the dividend. We subtract 2-2. It will turn out to be zero, so we take down the next number and again look for the quotient. We perform a mathematical operation until we get zero.
• After getting the final result, make a check using multiplication: 125x2=250.

It is advisable to teach a third-grader to reason out loud in the process of calculating, to perform actions on a draft. First, speak the algorithm together, then just listen to the student and help correct the mistakes.

On a note! Teach your child to constantly check himself. The student must understand that the value of the remainder of the subtraction in the division column must always be less than the divisor.

Division by a single number

Grab a piece of paper and a pen and have your child sit next to you. First, write down an example of a corner yourself. To divide by a single digit, choose numbers that give the result without a remainder (full answer).

The first lesson can be built like this:

1. Put a picture with a sample of division in a column in front of the child.
2. Come up with your own example. Let it be 254:2
3. The task must be written in a corner. Leave it to the student. He can see how the recording is made in the picture.
4. Ask a third grader: "What number should be divided by 2 first?". At this point, it is important to explain that the dividend must be equal to or greater than the divisor. The kid will select the first number from the given figure for division: 2 … 54
5. Now determine together how many twos will fit in the number 2. Answer: 1.
6. We write down the private under the corner.
7. Multiply 1 by 2 and write the result under the dividend.
8. Subtract.
9. Since it turned out to be 0, we demolish the next figure under the line after subtraction: 5.
10. Again we ask the question: “How many twos will fit in 5?” The kid remembers the multiplication table or selects the quotient using logic. Answers: 2.
11. Write 2 as a quotient, multiply by 2.
12. The result (4) is written under 5.
13. We take away.
14. It remains 1. One cannot be divided by 2, so we demolish the remainder of the dividend down. It turns out 14.
15. We divide 14 by 2. We write down in private 7.
16. Multiply by 2. Write under the line 14.
17. We take away.
18. The end result should always be 0.
19. As a result, the child will have the following record:

To consolidate, write down 3-5 more examples per division on the same piece of paper. Do not go far from the student, do not hide the sample, do not turn the lesson into a test. The baby is just learning to share. At this stage, help him, prompt and push him to the right decision to increase his self-confidence.

On a note! To automate the skill of dividing by a column, you can make a small memo where each stage of the mathematical action is written. Allow the student to look into it until he himself forgets about the sample.

Division by two digits

When a student of the 3rd grade has mastered the division by a single number, you can proceed to the next stage - working with double-digit numbers. Start with simple, clear examples so that the baby understands the algorithm of actions. For example, take the numbers 196 and 28 and explain the principle:

1. First, choose an approximate number for the answer. To do this, find out approximately how many digits of 28 will fit in 196. For convenience, you can round both numbers: 200:30. It will turn out no more than 6. The resulting number does not need to be written down, this is just a guess.
2. We check the result by multiplying: 28x6. It turns out 196. The assumptions turned out to be correct.
3. Write down the answer: 196:28 =6.

Another learning option: dividing by a two-digit number with a corner. This method is more suitable for working with numbers from four digits, that is, thousands. Here is a simple example:

1. Write on a piece of paper 4070, draw a corner and sign the divisor - 74.
2. Determine from what number you will start dividing. Ask your child if 4 can be divided by 74, 40? As a result, the baby will understand that first you need to limit yourself to the number 407. Outline the resulting figure from above in a semicircle. 0 will be left out.
3. Now we need to find out how many 74 will fit in 407. We act with the help of logic and multiplication check. It turns out 5. We write the result under the corner (under the divider).
4. Now we multiply 74 by 5 and write the result under the dividend. It will turn out 370. It is important to start recording from the first number on the left.
5. After recording, you need to draw a horizontal line and subtract 370 from 407. You get 37.
6. 37 cannot be divided by 74, so the remaining 0 in the top row is demolished.
7. Now we divide 370 by 74. We select the factor (5) and write it down under the corner.
8. We multiply 5 by 74, write the result in a column. Get 370.
9. Again we get the difference. The result will be 0. This means that the division is considered complete without a remainder. 4070:74=55. Private look at the corner.

To check the correctness of the solution, multiply: 74x55=4070.

There is an opinion! Many parents consider it unacceptable to have a solution book with GDZ in the house. But in vain. With the help of ready-made tasks, the child can easily test himself. The main thing is to correctly explain to the student the purpose of the collection of DZ with answers.

Multi-digit numbers

The most difficult tasks for children are tasks for three-digit and four-digit numbers. It is difficult for a fourth-grader to operate with thousands and hundreds of thousands. The student has the following problems:

1. Cannot determine the partial number of the dividend for the first action. Return to the study of the digits of natural numbers, work on the development of the baby's attention.
2. Skips 0 in private entry. This is the most common problem. As a result, the child gets a number a few digits less than the correct one. To avoid this error, you need to print a memo with a sequence of actions in examples where there are zeros in the middle of the quotient. Offer your child a simulator with such tasks to practice the skill.

When learning to solve problems with large numbers, proceed in stages:

1. Explain what an incomplete dividend is and why it should be singled out.
2. Practice finding the divisible verbally without further problem solving. For example, give the children the following tasks:

Find the incomplete quotient in the examples: 369:28; 897:12; 698:36.

1. Now proceed to the solution on paper. Write down in a column: 1068:89.
2. First you need to separate the incomplete dividend. You can use a comma above the numbers.

On a note! Examples with seven-digit numbers with third graders do not need to be solved. It's too much. It is enough to dwell on tasks with five-digit numbers (up to 10,000). The division of millions of children go through high school.

Division with remainder

The final stage of the lessons to consolidate the division skill will be the solution of tasks with a remainder. They will definitely meet in the solution book for the 3rd-4th grade. In gymnasiums with a mathematical bias, schoolchildren study not only incomplete numbers, but also decimal fractions. The form of writing an example with a corner will remain the same, only the answer will differ.

Take simple examples of division with a remainder, you can convert already solved tasks with an integer in the answer, adding one to the dividend. This is very convenient for the child, he will immediately see how the examples are similar and how they differ.

The lesson might look like this:

On a note! It is not necessary to separate an integer from the remainder of a comma, to make a fraction out of it at the initial stage of learning division. Record the remainder separately so that the student sees the end result of the difference in the column.

How to check

Division is checked by multiplication: the divisor is multiplied by the divisor. You can do this in a column:

Now let's check:

To check division with a remainder:

1. Multiply the total quotient by the divisor.
2. Add the remainder to the result.

34+1 (remainder) =35

The algorithm for checking the correctness of the solution of the division example does not change from the bit depth of the digits.

Important! At first, ask the child to paint the multiplication check in detail in order to check and consolidate knowledge of the table.

Examples for training

Training tasks help you learn how to quickly solve examples with division. Cards can end each lesson after passing a new topic.

unambiguous

Double digits

polysemantic

Download cards

As a home math simulator, use flashcards with examples. Include different cases in them: with single-digit and multi-digit numbers, division with a full result and a remainder. You can download the cards for free. Handout material must be printed for verification work.

Mistakes with division in children in elementary school are quite common. Give this topic maximum attention and time so that the assimilation of the subsequent material takes place without hesitation. Use flashcards, video tutorials, constant skill training and repetition of topics in a playful way. Then home lessons will not bore the child and will be held with maximum benefit.

IMPORTANT! *when copying article materials, be sure to indicate an active link to the first

Column division(you can also see the name division corner) is a standard procedure inarithmetic, designed to divide simple or complex multi-digit numbers by breakingdivision into a number of simpler steps. As in all division problems, a single number, calleddivisible, is divided into another, calleddivider, producing a result calledprivate.

A column can be used to divide both natural numbers without a remainder, and division of natural numbers with the rest.

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 whendivision of natural numbers by a column. Let's say right away that in writing to perform division by a columnit is most convenient on paper with a checkered line - so 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, after which between the writtennumbers represent the symbol of the form.

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

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

It can be seen from the above diagram that the desired quotient (or incomplete quotient when dividing 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 of the availability of space on the page in advance. In doing so, one should be guidedrule: the greater the difference in the number of characters in the records of the dividend and divisor, the morespace will be required.

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

How to divide into a column is best explained with an example.Calculate:

512:8=?

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

Their quotient (result) will be written under the divisor. Our number is 8.

1. We define an incomplete quotient. First, we look at the first digit from the left in the dividend entry.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 to the consideration the followingon the left, the digit in the record of the dividend, and work further with the number determined by the two considerednumbers. For convenience, we select in our record the number with which we will work.

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

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

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

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

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

4. Between 51 and 48 on the left, put "-" (minus). Subtract according to the rules of subtraction in column 48 and below the linewrite down the result.

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

The remainder turned out to be 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 there is a productcloser 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 figure located in the same column in the record of the dividend. If inthere are no digits in this column, then the division by a column ends here.

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

The remainder is zero. This means that the numbers are divided completely (without a remainder). If after the lastsubtracting zero, and there are no more digits left, then this is the remainder. We add it to the private inbrackets (e.g. 64(2)).

Division by a column of multivalued natural numbers.

Division by a natural multi-digit number is done in a similar way. At the same time, in the firstThe “intermediate” dividend includes so many high-order digits that it turns out to be more 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 the number made up of the digits of the three most significant digits - 197.
• The number 197 is greater than 26, divide 197 tens by 26: 197: 26 = 7 (15 tens left).
• We translate 15 tens into units, add 6 units from the category of units, we get 156.
• Divide 156 by 26 to get 6.

So 1976: 26 = 76.

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

Division with a decimal fraction in a quotient.

Decimal fractions online. Convert decimals to common fractions and common fractions to decimals.

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

For example, 64 divided by 5.

• Divide 6 tens by 5 to get 1 tens and 1 tens remainder.
• We translate the remaining ten into units, add 4 from the category of units, we get 14.
• 14 units divided by 5, we get 2 units and 4 units in the remainder.
• We translate 4 units into tenths, we get 40 tenths.
• Divide 40 tenths by 5 to get 8 tenths.

So 64:5 = 12.8

Thus, if when dividing a natural number by a natural one-digit or many-digit numberthe remainder is obtained, then you can put in a private comma, convert the remainder to the units of the next,smaller digit and continue dividing.

The division of natural numbers, especially multi-valued ones, is conveniently carried out by a special method, which is called division by a column (in a column). You can also see the name corner division. Immediately, we note that the column can be carried out both division of natural numbers without a remainder, and division of natural numbers with a remainder.

In this article, we will understand how division by a column is performed. Here we will talk about the writing rules, and about all intermediate calculations. First, let us dwell on the division of a multi-valued natural number by a single-digit number by a column. After that, we will focus on cases where both the dividend and the divisor are multi-valued natural numbers. The whole theory of this article is provided with characteristic 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 divide in a column in writing on paper with a checkered line - so there is less chance of going astray from the desired row and column.

First, the dividend and the 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 notation when divided into a column will be:

Take a look at the following diagram illustrating the places for writing the dividend, divisor, quotient, remainder, and intermediate calculations when dividing by a column.

From the above diagram, it can be seen that the desired 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 of the availability of space on the page in advance. In this case, one 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 is required. For example, when dividing a natural number 614,808 by 51,234 by a column (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 numbers 8 058 and 4 (here the difference in the number of characters is 4−1=3 ). To confirm our words, we present the completed records of division by a column of these natural numbers:

Now you can go directly to the process of dividing natural numbers by a column.

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

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 useful to practice the initial skills of division by a column on these simple examples.

Example.

Let us need to divide by a column 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 by a column.

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

Now we start to figure out how many times the divisor is in the dividend. To do this, we successively 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 down under the dividend, and in place of the private we write the number by which we multiplied the divisor. If we get a number greater than the divisible, 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 got a number equal to the dividend, so we write it under the dividend, and in place of the private we write the number 4. The record will then look like this:

The final stage of dividing single-digit natural numbers by a column remains. Under the number written under the dividend, you need to draw a horizontal line, and subtract numbers above this line in the same way as it is done when subtracting natural numbers with a column. The number obtained 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 a finished record of division by a column of the number 8 by 2. We see that the quotient 8:2 is 4 (and the remainder is 0 ).

Answer:

8:2=4 .

Now consider how the division by a column of single-digit natural numbers with a remainder is carried out.

Example.

Divide by 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 a 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 comparison of 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 (multiplication was carried out on 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.

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

Answer:

7:3=2 (rest. 1) .

Now you can move on to dividing multi-valued natural numbers by single-digit natural numbers by a column.

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

First, we look at the first digit from the left in the dividend entry. 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 the next digit to the left in the dividend record, and work further with the number determined by the two digits in question. For convenience, we select in our record the number with which we will work.

The first digit from the left in the dividend 140,288 is the number 1. The number 1 is less than the divisor 4, so we also look at the next digit on the left in the dividend record. At the same time, we see the number 14, with which we have to work further. We select this number in the notation of the dividend.

The following points 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 successively 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, then we write it under the selected number according to the notation rules used when subtracting by a column of natural numbers. The number by which the multiplication was carried out is written in place of the quotient during the first pass of the algorithm (during subsequent passes of 2-4 points of the algorithm, this number is written to the right of the numbers already there). When a number is obtained that is greater than the number x, then under the selected 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).

We multiply the divisor of 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>fourteen . Since at the last step we got the number 16, which is greater than 14, then under the selected number we write the number 12, which turned out at the penultimate step, and in place of the quotient we write the number 3, since in the penultimate paragraph the multiplication was carried out precisely on it.


At this stage, from the selected number, subtract the number below it in a column. Below the horizontal line is the result of the subtraction. However, if the result of the subtraction is zero, then it does not need to be written down (unless the subtraction at this point is the very last action that completely completes the division by a column). Here, for your control, it will not be superfluous to compare the result of subtraction with the divisor and make sure that it is less than the divisor. Otherwise, a mistake has been made somewhere.

We need to subtract the number 12 from the number 14 in a column (for the correct notation, you must not forget to put a minus sign to the left of the subtracted numbers). After the completion of this action, the number 2 appeared under the horizontal line. Now we check our calculations by comparing the resulting number with a divisor. Since the number 2 is less than the divisor 4, you can safely move on to the next item.


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 zero), we write down the number located in the same column in the record of the dividend. If there are no numbers in the record of the dividend in this column, then the division by a column ends here. After that, we select the number formed under the horizontal line, take it as a working number, and repeat with it from 2 to 4 points of the algorithm.

Under the horizontal line to the right of the number 2 already there, we write 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 the actions of the second, third and fourth points of the algorithm with it.

We multiply the divisor of 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 subtraction by a column. Since we subtract equal natural numbers, then, due to the property of subtracting equal natural numbers, we get zero as a result. We do not write zero (since this is not the final stage of dividing by a column), but we remember the place where we could write it down (for convenience, we will mark this place with a black rectangle).


Under the horizontal line to the right of the memorized place, we write the number 2, since it is she who is in the entry of the dividend 140 288 in this column. Thus, under the horizontal line we have the number 2 .


We take the number 2 as a working number, mark it, and once again we will have to perform the steps from 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 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 subtraction by 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, we add the number 8 (since it is in this column in the record of the dividend 140 288). Thus, under the horizontal line is the number 28.


We accept this number as a worker, mark it, and repeat steps 2-4 of paragraphs.

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

It remains for the last time to carry out the actions from points 2, 3, 4 (we provide it to you), after which you will get a complete picture of dividing natural numbers 140 288 and 4 in a column:

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

Thus, looking at the completed record of dividing the multi-valued natural number 140 288 by the single-valued natural number 4, we see that the number 35 072 is private (and the remainder of the division is zero, it is on 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 7136 and the divisor is a single natural number 9.

Solution.

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

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

Repeating the cycle, we will have

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

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

Answer:

7 136:9=792 (rest 8) .

And this example demonstrates how long division should look like.

Example.

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

Solution.

It is most convenient to perform division by a column.

Answer:

7 042 035:7=1 006 005 .

Division by a column of multivalued natural numbers

We hasten to please you: if you have well mastered the algorithm for dividing by a column from the previous paragraph of this article, then you already almost know how to perform division by a column of multivalued natural numbers. This is true, since steps 2 to 4 of the algorithm remain unchanged, and only minor changes appear in the first step.

At the first stage of dividing into a column of multi-valued natural numbers, you need to look not at the first digit on the left in the dividend entry, but at as many of them as there are digits in the divisor entry. 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 record of the dividend. After that, the actions indicated in paragraphs 2, 3 and 4 of the algorithm are performed until the final result is obtained.

It remains only to see the application of the algorithm for dividing by a column of multi-valued natural numbers in practice when solving examples.

Example.

Let's perform division by a column of multivalued natural numbers 5562 and 206.

Solution.

Since 3 characters are involved in the record of the divisor 206, we look at the first 3 digits on the left in the record of 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 proceed 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 the multiplication is difficult, then it is better to perform the multiplication of natural numbers in a column): 206 0=0<556 , 206·1=206<556 , 206·2=412<556 , 206·3=618>556 . Since we got a number that is greater than 556, then under the selected 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 it was multiplied at the penultimate step). The column division entry takes the following form:

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

Under the horizontal line to the right of the number available there, we write the number 2, since it is in the record of the dividend 5 562 in this column:

Now we work with the number 1442, select it, and go through steps two through four again.

We multiply the divisor 206 by 0 , 1 , 2 , 3 , ... until we 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 subtract by a column, we get zero, but we don’t write it down right away, but only remember its position, because we don’t know if the division ends here, or we will have to repeat the steps of the algorithm again:

Now we see that under the horizontal line to the right of the memorized position, we cannot write down any number, since there are no numbers in the record of the dividend in this column. Therefore, this division by a column is over, and we complete the entry:

• Mathematics. Any textbooks for grades 1, 2, 3, 4 of educational institutions.
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Division of multi-digit numbers is easiest to do in a column. Column division is also called corner division.

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

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

Under the horizontal line, the quotient resulting from the calculations will be written in stages:

Under the dividend, intermediate calculations will be written:

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

How to divide by a column

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

The division by a column is carried out in stages. The first thing we need to do is define the incomplete dividend. Look at the first digit of the dividend:

this number is 7, since it is less than the divisor, then we cannot start dividing from it, so we need to take one more digit from the dividend, the number 78 is greater than the divisor, so we start dividing from it:

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

Having determined the incomplete dividend, we can find out how many digits there will be in the private, 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, which means that the quotient will consist of 2 digits.

Having found out the number of digits that should turn out in a private one, you can put dots in its place. If, at the end of the division, the number of digits turned out to be more or less than the indicated points, then a mistake 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 successively multiply the divisor by natural numbers 1, 2, 3, ... until we get a number as close as possible to the incomplete divisible or equal to it, but not exceeding it. Thus, we get the number 6, write it under the divisor, and subtract 72 from 78 (according to the rules of column subtraction) (12 6 \u003d 72). After we subtracted 72 from 78, we got a remainder of 6:

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

To the resulting remainder - 6, we demolish the next digit of the dividend - 0. As a result, we got an incomplete dividend - 60. We determine how many times 12 is contained in the number 60. We get the number 5, write it into 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 that 780 is divided by 12 completely. As a result of performing division by a column, we found the quotient - it is written under the divisor:

Consider an example where zeros are obtained in the quotient. Let's say we need to divide 9027 by 9.

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

We demolish the next digit of the dividend - 0. We recall that when dividing zero by any number, there will be zero. We write to private zero (0: 9 = 0) and subtract 0 from 0 in intermediate calculations. Usually, in order not to pile up intermediate calculations, the calculation with zero is not written down:

We demolish 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, zero is written into the quotient and the next digit of the dividend is taken down:

We determine how many times 9 is contained in the number 27. We get the number 3, write it into 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:

Consider an example where 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 it into the quotient 5 and subtract 30 from 30. The remainder is zero. As already mentioned, it is not necessary to write down zero in the remainder in intermediate calculations:

We demolish the next digit of the dividend - 0. Since when dividing zero by any number there will be zero, we write it to private zero and subtract 0 from 0 in intermediate calculations:

We demolish the next digit of the dividend - 0. We write one more zero into the quotient and subtract 0 from 0 in intermediate calculations. at the very end of the calculation, it is usually written to show that the division is complete:

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

Division by a column with a remainder

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

We determine the incomplete dividend - this is the number 134. We write in the quotient 5 and subtract 115 from 134. The remainder turned out to be 19:

We demolish the next digit of the dividend - 0. Determine how many times 23 is contained in the number 190. We get the number 8, write it into a 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. Suppose we need to divide 3 by 10. We see that 10 is never contained in the number 3, so we write it to the quotient 0 and subtract 0 from 3 (10 0 = 0). We draw a horizontal line and write down the remainder - 3:

3: 10 = 0 (remainder 3)

Column Division Calculator

This calculator will help you perform division by a column. Just enter the dividend and divisor and click the Calculate button.

How to teach a child to divide? The simplest method is learn division by a column. This is much easier than doing mental calculations, it helps not to get confused, not to “lose” numbers and develop a mental scheme that will work automatically in the future.

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How is it carried out

Division with a remainder is a method in which a number cannot be divided into exactly several parts. As a result of this mathematical operation, in addition to the whole part, an indivisible piece remains.

Let's take a simple example how to divide with a remainder:

There is a can of 5 liters of water and 2 cans of 2 liters. When water is poured from a five-liter jar into a two-liter jar, 1 liter of unused water will remain in the five-liter jar. This is the remainder. Digitally it looks like this:

5:2=2 rest (1). Where is 1 from? 2x2=4, 5-4=1.

Now consider the order of division into a column with a remainder. This visually facilitates the calculation process and helps not to lose numbers.

The algorithm determines the location of all elements and the sequence of actions by which the calculation is performed. As an example, let's divide 17 by 5.

Main stages:

1. Correct entry. Divisible (17) - located on the left side. To the right of the dividend, write the divisor (5). A vertical line is drawn between them (indicates the sign of division), and then, from this line, a horizontal line is drawn, emphasizing the divisor. The main features are indicated in orange.
2. The search for the whole. Next, the first and simplest calculation is carried out - how many divisors fit in the dividend. Let's use the multiplication table and check in order: 5*1=5 - fits, 5*2=10 - fits, 5*3=15 - fits, 5*4=20 - doesn't fit. Five times four is more than seventeen, which means that the fourth five does not fit. Back to three. A 17 liter jar will fit 3 five liter jars. We write the result in the form: 3 we write under the line, under the divisor. 3 is an incomplete quotient.
3. Definition of the remainder. 3*5=15. 15 is written under the dividend. We draw a line (indicates the sign "="). Subtract the resulting number from the dividend: 17-15=2. We write the result below under the line - in a column (hence the name of the algorithm). 2 is the remainder.

Note! When dividing in this way, the remainder must always be less than the divisor.

When the divisor is greater than the dividend

There are cases when the divisor is greater than the dividend. Decimal fractions in the program for the 3rd grade have not yet been studied, but, following the logic, the answer must be written in the form of a fraction - at best a decimal, at worst a simple one. But (!) in addition to the program, the calculation method limits the task: it is necessary not to divide, but to find the remainder! some of them are not! How to solve such a problem?

Note! There is a rule for cases when the divisor is greater than the dividend: the incomplete quotient is 0, the remainder is equal to the dividend.

How to divide the number 5 by the number 6, highlighting the remainder? How many 6 liter jars will fit in a 5 liter jar? because 6 is greater than 5.

According to the task, it is necessary to fill 5 liters - not a single one is filled. So, all 5 are left. Answer: incomplete quotient = 0, remainder = 5.

Division begins to be studied in the third grade of school. By this time, students should already be, which allows them to divide two-digit numbers into single-digit ones.

Solve the problem: 18 sweets need to be distributed to five children. How many candies are left?

Examples:

Find the incomplete quotient: 3*1=3, 3*2=6, 3*3=9, 3*4=12, 3*5=15. 5 - bust. We return to 4.

Remainder: 3*4=12, 14-12=2.

Answer: incomplete quotient 4, 2 left.

You may ask why, when divided by 2, the remainder is either 1 or 0. According to the multiplication table, between digits that are multiples of two there is a difference per unit.

Another task: 3 pies must be divided into two.

Divide 4 pies between two.

Divide 5 pies between two.

Working with multi-digit numbers

The 4th grade program offers a more complex division process with an increase in calculated numbers. If in the third grade the calculations were carried out on the basis of the basic multiplication table ranging from 1 to 10, then the fourth-graders carry out calculations with multi-digit numbers over 100.

This action is most convenient to perform in a column, since the incomplete quotient will also be a two-digit number (in most cases), and the column algorithm facilitates calculations and makes them more visual.

Let's divide multi-digit numbers to two-digit: 386:25

This example differs from the previous ones in the number of calculation levels, although the calculations are carried out according to the same principle as before. Let's take a closer look:

386 is the dividend, 25 is the divisor. It is necessary to find the incomplete quotient and extract the remainder.

First level

The divisor is a two-digit number. The dividend is three-digit. We select the first two left digits from the dividend - this is 38. We compare them with the divisor. 38 over 25? Yes, so 38 can be divided by 25. How many whole 25s are in 38?

25*1=25, 25*2=50. 50 is greater than 38, go back one step.

Answer - 1. We write the unit to the zone not full private.

38-25=13. We write the number 13 under the line.

Second level

13 over 25? No - it means you can “lower” the number 6 down by adding it next to 13, on the right. It turned out 136. Is 136 more than 25? Yes, it means you can subtract it. How many times does 25 fit into 136?

25*1=25, 25*2=50, 25*3=75, 25*4=100, 25*5=125, 256*=150. 150 is greater than 136 - go back one step. We write the number 5 in the incomplete quotient zone, to the right of the unit.

We calculate the remainder:

136-125=11. We write under the line. 11 over 25? No, division is not possible. Does the dividend have digits left? No, there is nothing more to share. Calculations completed.

Answer: the incomplete quotient is 15, with a remainder of 11.

And if such a division is proposed, when the two-digit divisor is greater than the first two digits of the multi-valued dividend? In this case, the third (fourth, fifth and subsequent) digit of the dividend takes part in the calculations immediately.

Here are some examples division with three- and four-digit numbers:

75 is a two-digit number. 386 - three-digit. Compare the first two digits on the left with the divisor. 38 over 75? No, division is not possible. We take all 3 numbers. 386 over 75? Yes, division is possible. We carry out calculations.

75*1=75, 75*2=150, 75*3=225, 75*4=300, 75*5= 375, 75*6=450. 450 is greater than 386 - we go back a step. We write down 5 in the zone of incomplete quotient.

Find the remainder: 386-375=11. 11 over 75? No. Are there any digits left in the dividend? No. Calculations completed.

Answer: incomplete quotient \u003d 5, in the remainder - 11.

We check: 11 is greater than 35? No, division is not possible. We substitute the third number - is 119 greater than 35? Yes, we can take action.

35*1=35, 35*2=70, 35*3=105, 35*4=140. 140 is greater than 119 - we go back one step. We write 3 in the zone of incomplete balance.

Find the remainder: 119-105=14. 14 over 35? No. Are there any digits left in the dividend? No. Calculations completed.

Answer: incomplete quotient = 3, left - 14.

Checking if 11 is greater than 99? No - we substitute one more digit. 119 over 99? Yes, let's start the calculations.

11<99, 119>99.

99*1=99, 99*2=198 - bust. We write 1 in the incomplete quotient.

Find the remainder: 119-99=20. twenty<99. Опускаем 5. 205>99. We calculate.

99*1=99, 99*2=198, 99*3=297. Bust. We write 2 in the incomplete quotient.

Find the remainder: 205-198=7.

Answer: incomplete quotient = 12, remainder - 7.

Division with remainder - examples

Learning to divide in a column with a remainder

Output

This is how the calculations are done. If you are careful and follow the rules, then there will be nothing complicated here. Every student can learn to count with a column, because it is fast and convenient.