Column division examples. The secret of an experienced teacher: how to explain long division to a child

One of important stages in teaching a child mathematical operations - teaching division operations prime numbers. How to explain division to a child, when can you start mastering this topic?

In order to teach a child division, it is necessary that by the time he learns he has already mastered such mathematical operations, like addition, subtraction, and also had a clear understanding of the very essence of the operations of multiplication and division. That is, he must understand that division is the division of something into equal parts. It is also necessary to teach multiplication operations and learn the multiplication table.

I have already written about this. This article may be useful to you.

We master the operation of division (division) into parts in a playful way

At this stage, it is necessary to form in the child an understanding that division is the division of something into equal parts. The easiest way to teach a child this is to invite him to share a certain number of items among his friends or family members.

Let's say you take 8 identical cubes and ask your child to divide them into two equal parts - for him and for another person. Vary and complicate the task, invite the child to divide 8 cubes not between two, but into four people. Analyze the result with him. Change the components, try with a different number of objects and people into whom these objects need to be divided.

Important: Make sure that at first the child operates with an even number of objects, so that the result of division is the same number of parts. This will be useful at the next stage, when the child needs to understand that division is the inverse operation of multiplication.

Multiply and divide using the multiplication table

Explain to your child that in mathematics, the opposite of multiplication is called division. Using the multiplication table, demonstrate to the student the relationship between multiplication and division using any example.

Example: 4x2=8. Remind your child that the result of multiplication is the product of two numbers. After this, explain that division is the inverse of multiplication and illustrate this clearly.

Divide the resulting product “8” from the example by any of the factors “2” or “4”, and the result will always be a different factor that was not used in the operation.

You also need to teach the young student the names of the categories that describe the operation of division - “dividend”, “divisor” and “quotient”. Using an example, show which numbers are the dividend, divisor and quotient. Consolidate this knowledge, it is necessary for further training!

Essentially, you need to teach your child the multiplication table in reverse, and it is necessary to memorize it just as well as the multiplication table itself, because this will be necessary when you start learning long division.

Divide by column - let's give an example

Before starting the lesson, remember with your child what the numbers are called during the division operation. What is a “divisor”, “divisible”, “quotient”? Teach how to accurately and quickly identify these categories. This will be very useful when teaching your child how to divide prime numbers.

We explain clearly

Let's divide 938 by 7. In this example, 938 is the dividend, 7 is the divisor. The result will be a quotient, and that is what needs to be calculated.

Step 1. We write down the numbers, separating them with a “corner”.

Step 2. Show the student the numbers of the dividend and ask him to choose from them the smallest number that is greater than the divisor. Of the three numbers 9, 3 and 8, this number will be 9. Invite your child to analyze how many times the number 7 can be contained in the number 9? That's right, just once. Therefore, the first result we recorded will be 1.

Step 3. Let's move on to the design of division by column:

We multiply the divisor 7x1 and get 7. We write the resulting result under the first number of our dividend 938 and subtract it, as usual, in a column. That is, from 9 we subtract 7 and get 2.

We write down the result.

Step 4. The number we see less than divisor, so it needs to be increased. To do this, we combine it with the next unused number of our dividend - it will be 3. We assign 3 to the resulting number 2.

Step 5. Next we proceed according to known algorithm. Let's analyze how many times our divisor 7 is contained in the resulting number 23? That's right, three times. We fix the number 3 in the quotient. And the result of the product - 21 (7 * 3) is written below under the number 23 in a column.

Step.6 Now all that remains is to find the last number of our quotient. Using the already familiar algorithm, we continue to do calculations in the column. By subtracting in column (23-21) we get the difference. It equals 2.

From the dividend we have one number left unused - 8. We combine it with the number 2 obtained as a result of subtraction, we get - 28.

Step.7 Let's analyze how many times our divisor 7 is contained in the resulting number? That's right, 4 times. We write the resulting number into the result. So, we get the quotient obtained by dividing by a column = 134.

How to teach a child division - reinforcing the skill

The main reason why many schoolchildren have problems with mathematics is the inability to quickly do simple arithmetic calculations. And on this basis all mathematics is built. elementary school. Especially often the problem is in multiplication and division.
In order for a child to learn how to quickly and efficiently carry out division calculations in his head, it is necessary correct technique learning and consolidation of skills. To do this, we advise you to use today’s popular textbooks on learning division skills. Some are designed for children to study with their parents, others for independent work.

1. "Division. Level 3. Workbook» from the largest international center additional education Kumon
2. "Division. Level 4. Workbook" from Kumon
3. “Not Mental Arithmetic. Child education system fast multiplication and division. In 21 days. Notepad-simulator." from Sh. Akhmadulin - author of best-selling educational books

The most important thing when you teach a child long division is to master the algorithm, which, in general, is quite simple.

If a child is good at using the multiplication table and “reverse” division, he will not have any difficulties. However, it is very important to constantly practice the acquired skill. Don't stop there once you realize that your child has grasped the essence of the method.

In order to easily teach your child division operations you need:

• So that at the age of two or three years he masters the whole-part relationship. He must develop an understanding of the whole as an inseparable category and the perception of a separate part of the whole as an independent object. For example, a toy truck is a whole, and its body, wheels, doors are parts of this whole.
• So that in the younger school age the child could freely operate with addition and subtraction of numbers and understood the essence of the processes of multiplication and division.

In order for a child to enjoy mathematics, it is necessary to arouse his interest in mathematics and mathematical operations, not only during learning, but also in everyday situations.

Therefore, encourage and develop your child’s observation skills, draw analogies with mathematical operations (counting and division operations, analysis of “part-whole” relationships, etc.) during construction, games and observations of nature.

Teacher, child development center specialist
Druzhinina Elena
website specifically for the project

Video story for parents on how to correctly explain long division to a child:

Division multi-digit or multi-digit numbers are convenient to produce in writing in a column. Let's figure out how to do this. Let's start by dividing a multi-digit number by a single-digit number, and gradually increase the digit of the dividend.

So let's divide 354 on 2 . First, let's place these numbers as shown in the figure:

We place the dividend on the left, the divisor on the right, and the quotient will be written under the divisor.

Now we begin to divide the dividend by the divisor bitwise from left to right. We find first incomplete dividend, for this we take the first digit on the left, in our case 3, and compare it with the divisor.

3 more 2 , Means 3 and there is an incomplete dividend. We put a dot in the quotient and determine how many more digits will be in the quotient - the same number as remained in the dividend after selecting the incomplete dividend. In our case, the quotient has the same number of digits as the dividend, that is, the most significant digit will be hundreds:

In order to 3 divide by 2 remember the multiplication table by 2 and find the number, when multiplied by 2 we get the greatest product, which is less than 3.

2 × 1 = 2 (2< 3)

2 × 2 = 4 (4 > 3)

2 less 3 , A 4 more, which means we take the first example and the multiplier 1 .

Recording 1 to the quotient in place of the first point (in the hundreds place), and write the found product under the dividend:

Now we find the difference between the first incomplete dividend and the product of the found quotient and the divisor:

The resulting value is compared with the divisor. 15 more 2 , which means we have found the second incomplete dividend. To find the result of division 15 on 2 again remember the multiplication table 2 and find the greatest product that is less 15 :

2 × 7 = 14 (14< 15)

2 × 8 = 16 (16 > 15)

The required multiplier 7 , we write it as a quotient in place of the second point (in tens). We find the difference between the second incomplete dividend and the product of the found quotient and divisor:

We continue the division, why we find third incomplete dividend. We lower the next digit of the dividend:

We divide the incomplete dividend by 2, putting the resulting value in the category of units of the quotient. Let's check the correctness of the division:

2 × 7 = 14

We write the result of dividing the third incomplete dividend by the divisor into the quotient and find the difference:

The difference we got equal to zero, then the division is done Right.

Let's complicate the problem and give another example:

1020 ÷ 5

Let's write our example in a column and define the first incomplete quotient:

The thousands place of the dividend is 1 , compare with the divisor:

1 < 5

We add the hundreds place to the incomplete dividend and compare:

10 > 5 – we have found an incomplete dividend.

We divide 10 on 5 , we get 2 , write the result into the quotient. The difference between the incomplete dividend and the result of multiplying the divisor and the found quotient.

10 – 10 = 0

0 we do not write, we omit the next digit of the dividend – the tens digit:

We compare the second incomplete dividend with the divisor.

2 < 5

We should add one more digit to the incomplete dividend; for this we put in the quotient, on the tens digit 0 :

20 ÷ 5 = 4

We write the answer in the category of units of the quotient and check: we write the product under the second incomplete dividend and calculate the difference. We get 0 , Means example solved correctly.

And 2 more rules for dividing into a column:

1. If the dividend and divisor have zeros in the low-order digits, then before dividing they can be reduced, for example:

As many zeros in the low-order digit of the dividend we remove, we remove the same number of zeros in the low-order digits of the divisor.

2. If there are zeros left in the dividend after division, then they should be transferred to the quotient:

So, let’s formulate the sequence of actions when dividing into a column.

1. Place the dividend on the left and the divisor on the right. We remember that we divide the dividend by isolating incomplete dividends bit by bit and dividing them sequentially by the divisor. The digits in the incomplete dividend are allocated from left to right from high to low.
2. If the dividend and divisor have zeros in the lower digits, then they can be reduced before dividing.
3. We determine the first incomplete divisor:

A) select the highest digit of the dividend into the incomplete divisor;

b) compare the incomplete dividend with the divisor; if the divisor is larger, then go to point (V), if less, then we have found an incomplete dividend and can move on to point 4 ;

V) add the next digit to the incomplete dividend and go to point (b).

1. We determine how many digits there will be in the quotient, and put as many dots in place of the quotient (under the divisor) as there will be digits in it. One point (one digit) for the entire first incomplete dividend and the remaining points (digits) are the same as the number of digits left in the dividend after selecting the incomplete dividend.
2. We divide the incomplete dividend by the divisor; to do this, we find a number that, when multiplied by the divisor, would result in a number either equal to the incomplete dividend or less than it.
3. We write the found number in place of the next quotient digit (dot), and write the result of multiplying it by the divisor under the incomplete dividend and find their difference.
4. If the difference found is less than or equal to the incomplete dividend, then we have correctly divided the incomplete dividend by the divisor.
5. If there are still digits left in the dividend, then we continue division, otherwise we go to point 10 .
6. We lower the next digit of the dividend to the difference and get the next incomplete dividend:

a) compare the incomplete dividend with the divisor, if the divisor is greater, then go to point (b), if less, then we have found the incomplete dividend and can proceed to point 4;

b) add the next digit of the dividend to the incomplete dividend, and write 0 in the place of the next digit (dot) in the quotient;

c) go to point (a).

10. If we performed division without a remainder and the last difference found is equal to 0 then we did the division correctly.

We talked about dividing a multi-digit number by a single-digit number. In the case where the divider is larger, division is performed in the same way:

Single-digit natural numbers are easy to divide in your head. But how to divide multi-digit numbers? If a number already has more than two digits, mental counting can take a lot of time, and the likelihood of errors when operating with multi-digit numbers increases.

Column division is a convenient method often used for dividing multi-digit natural numbers. It is this method that this article is devoted to. Below we will look at how to perform long division. First, let's look at the algorithm for dividing a multi-digit number by a single-digit number into a column, and then - multi-digit by multi-digit number. In addition to theory, the article provides practical examples of long division.

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It is most convenient to keep notes on squared paper, since when making calculations, the lines will prevent you from getting confused in the digits. First, the dividend and divisor are written from left to right in one line, and then separated by a special division sign in a column, which looks like:

Let's say we need to divide 6105 by 55, let's write:

We will write intermediate calculations under the dividend, and the result will be written under the divisor. In general, the column division scheme looks like this:

Please remember that calculations will require free space on the page. Moreover, than more difference in the dividend and divisor digits, the more calculations there will be.

For example, to divide the numbers 614,808 and 51,234 you will need less space, than for dividing the number 8058 by 4. Despite the fact that in the second case the numbers are smaller, the difference in the number of their digits is greater, and the calculations will be more cumbersome. Let's illustrate this:

It is most convenient to practice practical skills on simple examples. Therefore, let's divide the numbers 8 and 2 into a column. Of course, this operation is easy to perform in your head or using the multiplication table, but detailed analysis It will be useful for clarity, although we already know that 8 ÷ 2 = 4.

So, first we write down the dividend and divisor according to the column division method.

The next step is to find out how many divisors the dividend contains. How to do this? We successively multiply the divisor by 0, 1, 2, 3. . We do this until the result is a number equal to or greater than the dividend. If the result immediately results in a number equal to the dividend, then under the divisor we write the number by which the divisor was multiplied.

Otherwise, when we get a number greater than the dividend, under the divisor we write the number calculated at the penultimate step. In place of the incomplete quotient we write the number by which the divisor was multiplied at the penultimate step.

Let's go back to the example.

2 · 0 = 0 ; 2 · 1 = 2 ; 2 · 2 = 4 ; 2 · 3 = 6 ; 2 4 = 8

So, we immediately got a number equal to the dividend. We write it under the dividend, and write the number 4, by which we multiplied the divisor, in the place of the quotient.

Now all that remains is to subtract the numbers under the divisor (also using the column method). In our case, 8 - 8 = 0.

This example is dividing numbers without a remainder. The number obtained after subtraction is the remainder of the division. If it is equal to zero, then the numbers are divided without a remainder.

Now let's look at an example where numbers are divided with a remainder. Let's divide natural number 7 to the natural number 3.

In this case, sequentially multiplying three by 0, 1, 2, 3. . we get as a result:

3 0 = 0< 7 ; 3 · 1 = 3 < 7 ; 3 · 2 = 6 < 7 ; 3 · 3 = 9 > 7

Under the dividend we write the number obtained in the penultimate step. Using the divisor we write down the number 2 - the incomplete quotient obtained in the penultimate step. It was by two that we multiplied the divisor when we got 6.

To complete the operation, subtract 6 from 7 and get:

This example is dividing numbers with a remainder. The partial quotient is 2 and the remainder is 1.

Now, after considering elementary examples, let's move on to dividing multi-digit natural numbers into single-digit ones.

We will consider the column division algorithm using the example of dividing the multi-digit number 140288 by the number 4. Let us say right away that it is much easier to understand the essence of the method using practical examples, and this example was not chosen by chance, as it illustrates all the possible nuances of dividing natural numbers in a column.

1. Write the numbers together with the division symbol in a column. Now look at the first digit on the left in the dividend notation. Two cases are possible: the number defined by this digit is greater than the divisor, and vice versa. In the first case, we work with this number, in the second, we additionally take the next digit in the dividend notation and work with the corresponding two-digit number. In accordance with this point, let’s highlight in the example record the number with which we will work initially. This number is 14 because the first digit of the dividend 1 is less than the divisor 4.

2. Determine how many times the numerator is contained in the resulting number. Let's denote this number as x = 14. We successively multiply the divisor 4 by each member of the series of natural numbers ℕ, including zero: 0, 1, 2, 3 and so on. We do this until we get x or a number greater than x as a result. When the result of multiplication is the number 14, we write it under the highlighted number according to the rules for writing subtraction in a column. The factor by which the divisor was multiplied is written under the divisor. If the result of multiplication is a number greater than x, then under the highlighted number we write the number obtained at the penultimate step, and in place of the incomplete quotient (under the divisor) we write the factor by which the multiplication was carried out at the penultimate step.

In accordance with the algorithm we have:

4 0 = 0< 14 ; 4 · 1 = 4 < 14 ; 4 · 2 = 8 < 14 ; 4 · 3 = 12 < 14 ; 4 · 4 = 16 > 14 .

Under the highlighted number we write the number 12 obtained in the penultimate step. In place of the quotient we write the factor 3.

3. Subtract 12 from 14 using a column, write the result under the horizontal line. By analogy with the first point, we compare the resulting number with the divisor.

4. Number 2 less number 4, therefore we write down under the horizontal line after the two the number located in the next digit of the dividend. If there are no more digits in the dividend, then the division operation ends. In our example, after the number 2 obtained in the previous paragraph, we write down the next digit of the dividend - 0. As a result, we note a new working number - 20.

Important!

Points 2 - 4 are repeated cyclically until the end of the operation of dividing natural numbers by a column.

2. Let's count again how many divisors are contained in the number 20. Multiplying 4 by 0, 1, 2, 3. . we get:

Since we received a number equal to 20 as a result, we write it under the marked number, and in place of the quotient, in the next digit, we write 5 - the multiplier by which the multiplication was carried out.

3. We carry out the subtraction in a column. Since the numbers are equal, the result is the number zero: 20 - 20 = 0.

4. We will not write down the number zero, since this stage- not the end of division yet. Let’s just remember the place where we could write it down and write next to it the number from the next digit of the dividend. In our case, the number is 2.

We take this number as a working number and again carry out the steps of the algorithm.

2. Multiply the divisor by 0, 1, 2, 3. . and compare the result with the marked number.

4 0 = 0< 2 ; 4 · 1 = 4 > 2

Accordingly, under the marked number we write the number 0, and under the divisor in the next digit of the quotient we also write 0.

3. Perform the subtraction operation and write the result under the line.

4. To the right under the line add the number 8, since this is the next digit of the number being divided.

Thus, we get a new working number - 28. We repeat the points of the algorithm again.

Having done everything according to the rules, we get the result:

Move it below the line last digit dividend - 8. IN last time We repeat algorithm points 2 - 4 and get:

In the very bottom line we write the number 0. This number is written only at the last stage of division, when the operation is completed.

Thus, the result of dividing the number 140228 by 4 is the number 35072. This example is analyzed in great detail, and when solving practical tasks There is no need to describe all the actions so thoroughly.

We will give other examples of dividing numbers into a column and examples of writing solutions.

Example 1. Column division of natural numbers

Divide the natural number 7136 by the natural number 9.

After the second, third and fourth steps of the algorithm, the record will take the form:

Let's repeat the cycle:

The last pass, and we read the result:

Answer: The partial quotient of 7136 and 9 is 792 and the remainder is 8.

When solving practical examples, it is ideal not to use explanations in the form of verbal comments at all.

Example 2. Dividing natural numbers into a column

Divide the number 7042035 by 7.

Answer: 1006005

The algorithm for dividing multi-digit numbers into a column is very similar to the previously discussed algorithm for dividing a multi-digit number by a single-digit number. To be more precise, the changes concern only the first point, while points 2 - 4 remain unchanged.
If, when dividing by a single-digit number, we looked only at the first digit of the dividend, now we will look at as many digits as there are in the divisor. When the number determined by these digits is greater than the divisor, we take it as the working number. Otherwise, we add another digit from the next digit of the dividend. Then we follow the steps of the algorithm described above.

Let's consider the application of the algorithm for dividing multi-digit numbers using an example.

Example 3. Dividing natural numbers into a column

Let's divide 5562 by 206.

The divisor contains three signs, so let’s immediately select the number 556 in the dividend.
556 > 206, so we take this number as a working number and move on to point 2 of the agloritm.
Multiply 206 by 0, 1, 2, 3. . and we get:

206 0 = 0< 556 ; 206 · 1 = 206 < 556 ; 206 · 2 = 412 < 556 ; 206 · 3 = 618 > 556

618 > 556, so under the divisor we write the result of the penultimate action, and under the dividend we write the factor 2

Perform column subtraction

As a result of subtraction we have the number 144. To the right of the result, under the line, we write the number from the corresponding digit of the dividend and get a new working number - 1442.

We repeat points 2 - 4 with him. We get:

206 5 = 1030< 1442 ; 206 · 6 = 1236 < 1442 ; 206 · 7 = 1442

Under the marked working number we write 1442, and in the next digit of the quotient we write the number 7 - the multiplier.

We carry out subtraction into a column, and we understand that this is the end of the division operation: there are no more digits in the divisor to write to the right of the subtraction result.

To conclude this topic, we will give another example of dividing multi-digit numbers into a column, without explanation.

Example 5. Column division of natural numbers

Divide the natural number 238079 by 34.

Answer: 7002

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Algorithm for dividing numbers into a column, teaching a child. Features of division of multi-digit numbers and polynomials.

School gives the child not only discipline, development of talents and communication skills, but also knowledge of fundamental sciences. One of them is mathematics.

Although the program and the workload for students often change, dividing numbers with different numbers of digits into a column remains an unapproachable peak for many of them from the first attempt. Therefore, it is often impossible to do without training at home with your parents.

In order not to waste time and prevent the child from forming a lump of incomprehensibility in mathematics, refresh your memory of your knowledge of dividing numbers in a column. This article will help you with this.

How to correctly divide numbers into a column: division algorithm

To divide numbers in a column, follow these steps:

• write down the division action correctly on paper. Select the upper right corner of the notebook/sheet. If you are just learning to do long division, use squared paper. This way you will maintain the visual consistency of the solution,
• Line the space between the dividend and the divisor.
  The diagram below will help you.

• plan space for dividing into columns. The longer the number that is to be divided, and the larger the divisor, the lower the solution goes down on the page,
• Perform the first division operation with the number of digits of the dividend that is equal to the divisor. For example, if you have a single-digit number to the right of the dividing line, then consider the first one of the dividend; if it’s a two-digit number, then consider the first 2,
• multiply the numbers below and above the line and write the result under the numbers of the dividend that you indicated for the first action,
• complete the activity by subtracting and finding the remainder. Draw a horizontal line above it to separate the first step of the solution,
• add the next digit of the dividend to the remainder and continue solving,
• The last step of division is when you get 0 or a number less than the divisor from subtraction. In the second case, your answer will have a remainder, for example, 17 and 3 as a remainder.

How to explain division to a child and teach how to divide using a column?

First, consider a number of introductory factors:

• the child knows the multiplication table
• is well versed and able to apply in practice the operations of subtraction and addition
• understands the difference between the whole and its constituent elements

• play with the multiplication table. Place it in front of the child and show with examples how easy it is to use when dividing,
• Explain the location of the dividend, divisor, quotient, remainder. Invite your child to repeat these categories,
• turn the process into a game, come up with a story about numbers and division,
• Prepare visual aids for teaching. Counting sticks, apples, coins, toys, peeled notes or oranges will do. Offer to distribute them among different numbers of people, for example, between mom, dad and child,
• be the first to show your child what to do with even numbers, so that he sees the result of division, a multiple of two.

The process of mastering long division:

• write down the numbers, separating them with boundaries. Repeat with your child the arrangement of division categories,
• Invite him to analyze the digits of the dividend into a “greater-than-less” divisor. Help with a question - how many times one number is placed in the second. As a result, the child should select the number/numbers that he will use to perform the first action,
• Tell me the algorithm for determining the bit depth of a quotient. It is convenient to depict it with dots, which will then turn into numbers,
• Help to correctly identify and write the first number as a quotient, multiply it by the divisor, write the result under the dividend, and subtract. Explain that the result of a subtraction must always be less than the divisor. Otherwise, the action was performed with an error and should be redone,
• the next step is to analyze the situation with adding the second number from the dividend and determining the number of times the divisor is repeated in it,
• again help with recording the action,
• continue until the result of the difference is zero. This is only relevant for dividing numbers without a remainder,
• Reinforce your child’s knowledge with a few more examples. Make sure he doesn't get tired, otherwise give him a break.

How to divide a two-digit number into a single-digit and two-digit number in writing: examples, explanation

Let's start with a step-by-step analysis of examples of long division.

Perform the action on the numbers 25 and 2:

• write them down side by side and separate them with border lines,
• determine the required number of digits of the dividend for the first action,
• write the value under the divisor and the result of the multiplication under the dividend,
• do the subtraction,
• Add the second digit of the dividend and repeat the multiplication and subtraction steps.

For a partially completed task on dividing a two-digit number by a single-digit number by a column, see below:

Please note that dividing a two-digit number by a single-digit number with a column is possible in one step.

Second example. Divide 87 by 26 in a column.

The algorithm is similar to that discussed above with the only difference that you need to take into account 2 numbers of the divisor at once when determining the number of times the dividend is repeated.

To make the task easier for a child who is just learning the basics of division, ask him to focus on the first digits of the dividend and divisor. For example, 8:2=4. Have your child put this number under the line and do the multiplication. He needs to see with his own eyes that 4 is a lot and he needs to try with three.

Below is an example of a column dividing a two-digit number by a two-digit number with a remainder.

Third example. How to divide a number into a column with a zero in the answer.

First, we divide 15 by 15, the remainder is 0, the answer is 1. We take away 6, but it is not divisible by 15, so we put 0 in the answer. Next, 15 multiplied by 0 will be zero and subtract it from 6. We take away zero, which is at the end of the number, we get 60, which is divided by 15 and put 4 in response.

How to divide a three-digit number into a single-digit, two-digit and three-digit number: examples, explanation

Let's continue the analysis of the action of division by a column using examples with a three-digit dividend.

When the divisor is a single-digit number, the operation algorithm is similar to those discussed above.

Schematically it looks like this:

In the case of dividing a three-digit dividend by a two-digit divisor, choose with your child a number corresponding to the number of spaces of the second in the first part of the first or in general. That is, consider first the 2 digits of the three-digit dividend; if they are less than the divisor, then all three.

When your child has just begun to master long division, tell him how to perform actions with single-digit numbers. That is, with the first ones in the dividend and divisor. Let the child make a mistake that will lead to negative value subtraction and will return to selecting the number under the line, which will confuse the action immediately for the two-digit divisor.

Scheme for dividing three-digit by double digit numbers like this:

Three-digit values ​​in the divisor and dividend look cumbersome and scary for a child. Reassure him by explaining that the principle of operation is identical to that of dividing prime numbers.

The method of enumerating one digit at a time will help your child figure out each number separately. Only he will need more time for this action than in previous examples. For better visual perception, combine with arcs the number of numbers that will participate in the first action.

Diagram for dividing a three-digit number by a three-digit number.

How to divide four-digit, multi-digit large numbers, polynomials into polynomials: examples, explanation

In the case of dividing a four-digit number by any number that contains up to 4 orders of magnitude at the same time, pay the child’s attention to the nuances:

• Determining the correct number of orders after the division action. For example, in the example 6734:56 you should get a two-digit integer in the “quotient” column, and in the example 8956:1243 - a single-digit integer,
• the appearance of zeros in the quotient. When, during the solution, when carrying over the next number of the dividend, the result turns out to be less than the divisor,
• checking the result obtained by performing a multiplication operation. This nuance is relevant for dividing large numbers without a remainder. If the latter is present, then advise the child to check himself and divide the numbers into a column again.

Below is an example solution.

For large multi-digit numbers that are divisible into specific values ​​less than or equal to them in the number of digits, all the algorithms discussed above are relevant.

The child should be especially careful in such cases and correctly determine:

• the number of characters of the quotient, that is, the result
• digits of the dividend for the first action
• correctness of transfer of remaining numbers

Examples detailed solution below.

When performing division operations on polynomials, draw children's attention to a number of features:

• an action may or may not have a remainder. In the first case, write it in the numerator and the divisor in the denominator,
• to perform the subtraction operation, add the missing powers of the function multiplied by zero to the polynomial,
• transform polynomials by isolating repeating bi-/polynomials. Then reduce them and you will get the result without a trace.

Below row detailed examples with solutions.

How to divide with a remainder?

The algorithm for long division with a remainder is similar to the classic one. The only difference is the appearance of a remainder, which is less than the divisor. This means that the first one remains unchanged.

Write it down in your answer either:

• like a fraction, where the numerator is the remainder and the denominator is the divisor
• in words, for example, 73 whole and 6 remainder

How to divide decimal fractions with a comma?

There are several features of this division. If you perform an action with:

• a decimal fraction-dividend and an integer divisor, then proceed according to the usual algorithm until the dividend runs out of digits before the decimal point. Then put it in the quotient and continue moving the numbers until the end of the division,
• a number that is divisible by 10, 100, 100, etc., then move the comma in the dividend to the left by a number of digits equal to the number of zeros of the divisor. For example, 749.5:100=7.495,
• decimal fractions in both the divisor and the dividend at the same time, then first get rid of the comma from the second element. To do this, move it to the right in both fractional numbers by the number of digits that are separated from the divisor. For example, convert 416.788:5.3 to 4167.88:53 and do regular division in a column.

How to divide a smaller number by a larger number using a column?

With this division, your quotient will start at 0 and have a comma after it.

To help your child better understand this division and not get confused about the number of zeros and where the comma is placed in the quotient, give him the following example:

• carry out the first subtraction operation with zeros, written one at a time under the divisor and in the “quotient” column,
• put a comma in the quotient, and add a zero for the remainder after the difference and continue the usual long division,
• when the remainder of the subtraction is again less than the divisor, add a zero to the first and continue the action. The final result is getting a zero from the difference between the upper and lower numbers or repeating the remainder. In the latter case, there is a value in the period, that is, an infinitely repeating number/numbers.

Below is an example.

How to divide numbers with zeros using a column?

The sequence and algorithm of actions is similar to the classic one discussed in the first section.

Among the nuances we note:

• If there are zeros at the end of the divisor and dividend, feel free to reduce them. Invite your child to cross them out with a pencil and continue dividing as usual. For example, in a situation of 1200:400, a child can remove both zeros from both numbers, but in a situation of 15600:560 - only one extreme one,
• if zero is only in the divisor, then select the first digit for the action, focusing on the number in front of it. For example, in the example 6537:70, put 9 in the quotient as the first number. For this example multiply by both digits of the divisor and sign them under the three of the dividend.

When the dividend has a lot of zeros and the division process is over before you have used them all, then move them to the quotient after the numbers that were formed before. Example, 1000:2=500 - you moved the last two zeros.

So, we looked at the basic situations of dividing numbers different quantities digits in a column, determined the algorithm of action and emphasis for teaching the child.

Practice the knowledge you have acquired and help your child master mathematics.

Video: how to correctly divide numbers into a column?