How to multiply two-digit columns correctly. The secret of an experienced teacher: how to explain long division to a child

Unfortunately, children nowadays practically do not know how to do mental calculations. This happened due to the fact that modern technologies offer each child to solve the problem with a couple of clicks. For many children, the Internet has replaced not only textbooks, but also certain skills. You can increasingly hear from the younger generation that it is not at all necessary to know mathematics, since you always have a calculator or phone at hand. But the true significance of this science lies in the development of thinking, and not in overcoming the fear of being deceived by a trader in the market.

Long division helps elementary school students become familiar with number operations. Thanks to it, the multiplication table is fixed in memory, and the skill of performing addition and subtraction operations is honed.

To implement this arithmetic operation, you need to become familiar with its components:

1. Dividend - a number that is divided.

2. Divisor - the number that is divided by.

3. Quotient - the result obtained by division.

4. Remainder is the part of the dividend that cannot be divided.

American and European division models

The rules for long division are the same in all countries. There is only a difference in the graphic part, that is, in its recording. In the European system, the dividing line, or the so-called corner, is placed on the right side of the number being divided. The divisor is written above the corner line, and the quotient is written below the horizontal line of the corner.

Dividing into a column according to the American model involves placing a corner on the left side. The quotient is written above the horizontal line of the angle, directly above the number being divided. The divisor is written under the horizontal line, to the left of the vertical line. The process of performing the action itself does not differ from the European model.

Divide by a two-digit number

To use a two-digit value, you need to write it down according to the diagram, and then carry out the action. Column division begins with the highest digits of the number being divided. The first two digits are taken if the number formed by them is greater in value than the divisor. Otherwise, the first three digits are separated. The number they form is divided by the divisor, the remainder goes down, and the result is written in the dividing corner. After this, the digit from the next digit of the number being divided is transferred, and the procedure is repeated. This continues until the number is completely divided.

If it is necessary to divide a number with a remainder, it is written separately. If you need to completely divide a number, then after the end of the digits of the number a comma is placed in the answer, indicating the beginning of the fractional part, and instead of the digits, a zero is moved down each time.

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

Learning long division in the form of a game

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

You can set tasks this way:

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

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

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

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

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

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

Learning long division using the multiplication table

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

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

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

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

Learning long division in a notebook

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

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

The mathematical operation needs to be explained in detail:

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

Parents need to explain that this division has no remainder.

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

Learning division with remainder

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

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

Simple examples for a child

We can continue with the same example:

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

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

Teaching division through games

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

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

Learning to divide by column the smallest number by the largest

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

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

Division by a two-digit number is a complex operation that requires trained memory to remember initial and intermediate information.

As in other sections, start by practicing the simplest exercises, while simultaneously mastering more complex ones.

Division technique

When doing oral division, memorize numbers in pairs of digits, for example, 3542 as “thirty-five - forty-two.”

If the dividend is four-digit, then first determine the number of hundreds in the answer by dividing the first pair of digits by the divisor. Then work with the remainder of this division and the second pair. For example, when dividing 3542 by 11, the number of hundreds in the answer is 3, and dividing 242 by 11 gives 22, that is, the answer is 322.

Methods of division for various combinations of numbers are given in the following examples.

At the first stage, do not pay attention to division remainders - in practice, an approximate answer is usually sufficient.

In all examples in parentheses The remainder of the division is shown.

Division by 11-19

A.1. Multiply up to 19x9.

Division is the inverse operation of multiplication. Memorize the multiplication table up to 19×9 - this will allow you to quickly divide by numbers less than 20. Use this example to practice:

× =

A.2. Dividing a two-digit number.

Calculate the integer part and remainder:

: =

A.3. Division by 11.

: =

Dividing by 11 is easiest to do in the usual way, “in a column.”

When dividing a four-digit number, first determine the number of hundreds in the answer by dividing the first two digits of the number by 11. Then work with the remainder and the second pair of digits.
It's useful to remember that 1001 = 7 × 11 × 13 = 91 × 11. For example, dividing 1023 by 11 immediately results in 93.

You can learn to divide three-digit numbers by 11 right away if you remember the rule for multiplying a two-digit number by 11. For example:

577: 11 = 52 (5). You can immediately see that 572 is divided by 11 (5 + 2 = 7) and gives 52.
642: 11 = 58 (4). It is immediately clear that 638 is divided by even 11 and gives 58 (5 + 8 = 13).

A.4. Divide by 13.

: =

When dividing by 13 it is useful to remember:

1001 = 7 × 11 × 13 = 77 × 13.
104 = 8 × 13.

Algorithm for dividing by 13 using the number 6357 as an example:

First, let's use the fact that 1001 = 7 × 11 × 13. So, 6006: 13 = 42 × 11 = 462 (use the rule of multiplication by 11).
Next, you need to divide 357 − 6 = 351 by 13. Since 104 = 8 × 13, then 312: 13 = 24.
All that remains is to divide 351 − 312 = 39 by 13, which gives 3.
Adding it up, we get the answer: 489.

Sometimes it is easier to divide in the usual way, “in a column”, for example, 5265: 13 = 405, since 52: 13 = 4, 65: 13 = 5.

A.5. Divide by 15.

: =

When dividing by 15:

Determine the number of hundreds in your answer by dividing the first two digits of a four-digit number by 15.
Multiply the remaining number by 2, then divide by 30.

A.6. Divide by 17.

: =

When dividing by 17 it is useful to remember:

102 = 6 × 17.
1020 = 60 × 17.
1003 = 59 × 17.

Algorithm for dividing by 17 using the number 4493 as an example:

First, let's determine the number of hundreds in the answer: 44: 17 = 2 (10).
When dividing 1093 by 17, we use the fact that 1020: 17 = 60, and 73: 17 = 4 (5).
Adding it up, we get the answer: 264 (5).

Sometimes it is easier to divide in the usual way “in a column”, for example, 3572: 17 = 210 (2), since 34: 17 = 2, 172: 17 = 10 (2).

A.7. Divide by 19.

: =

When dividing by 19, it is useful to remember: 100: 19 = 5 (5).

Algorithm for dividing by 19 using the number 4126 as an example:

First, let's determine the number of hundreds in the answer: 41: 19 = 2 (3).
To divide 326 by 19, we use the fact that 100: 19 = 5 (5), so 300: 19 = 15 (15), and 41: 19 = 2 (3). So, 326: 19 = 17 (3).
Adding it up, we get the answer: 217 (3).

Sometimes it is easier to divide in the usual way “in a column”, for example, 1938: 19 = 102.

A.8. Divide by 12, 14, 16, 18.

: =

When dividing by an even number, first determine the number of hundreds in the answer by dividing the first two digits of the four-digit number by the divisor.

For the remaining number, either reduce the dividend and the divisor by 2 and then divide by a single digit number, or use the properties:

96 = 8 × 12.
96 = 6 × 16.
98 = 49 × 2 = 7 × 14.
90 = 18 × 5.

2149: 12 = 1 (hundred) + 9 × 8 + (9 × 4 + 49)/12 = 179 (1).
2149: 18 = 1 (hundred) + 3 × 5 + (3 × 10 + 49)/18 = 119 (7).

Division by 21-99

B.1. Divide by 91-99.

: =

To a first approximation, the answer is the number of hundreds in the dividend (45).
The number 100 is greater than 94 by 6. To calculate the next approximation, multiply the number of hundreds of the dividend by 6 and add the last two digits: 45 × 6 + 35 = 305.
Divide it by 94 in the same way: 305: 94 = 3 (3x6+5) = 3 (23).
Add up the answers. Total: 4535: 94 = 48 and 23/94.

Sometimes it is convenient to divide by 89 in the same way (since it is easy to multiply by 11 in intermediate calculations).

B.2. Division by numbers ending in 9.

: =

In this case, it is also convenient to use the rounding method. For example, you need to divide 3426 by 29.

Round the divisor up (from 29 we get 30).
Divide by 30 and calculate the remainder: 3426: 30 = 114 (6). This already gives an approximate answer - approximately 114.
To calculate the next approximation, add the answer and the remainder: 114 + 6 = 120.
Divide by 30 and calculate the remainder: 120: 30 = 4 (0). Thus, the integer part of the answer is equal to 114 + 4 = 118. And the remainder is equal to the sum of the last answer (4) with the last remainder (0), that is, 4. Total: 3426: 29 = 118 and 4/29.

B.3. Division by numbers ending in 7 and 8.

: =

The rounding method can also be used in this case.

Example of dividing 6742 by 48 by rounding (to 50):

First approximation: 67 × 2 = 134.
New dividend: 134 × 2 + 42 = 310.
Second approximation: 134 + 6 = 140 (the number 6 is 300:5).
Remainder: 6 × 2 + 10 = 22.
Answer: 6742: 48 = 140 (22).

As you master the method, you can also use it when dividing by numbers ending in 5 and 6 (which is more difficult, since it requires multiplying by 5 and 4 in intermediate calculations).

B.4. Division by numbers that are multiples of 11.

: =

When dividing by multiples of 11:

If the dividend is four digits, first determine the number of hundreds in the answer. To do this, divide the first pair of digits of the dividend by the divisor. Then work with the remainder of this division and the second pair.
Reduce the numerator and denominator by 11. This is usually not difficult, since dividing by 11 is easy and reduces the dividend by one place. If the dividend is not divisible by 11, discard a few units from it, which can then be added to the remainder.
Next, divide by the remaining factor of the original divisor.

When dividing by 33, it is sometimes more convenient to multiply the dividend and divisor by 3. Then the number of hundreds in the new divisor immediately gives an approximate answer.

Example 1. Divide 4359 by 33.

First, we determine the number of hundreds in the answer: 43: 33 = 1 (10). Next we work with the number 1059.
Let's multiply the dividend and divisor by 3: 1059: 33 = 3177: 99. The first approximation is equal to the number of hundreds in the new divisor: 31. The remainder is 31 + 77 = 108. Thus, 3177: 99 = 32 and 9/99.
Answer: 132 and 3/33 (the remainder is reduced to the original divisor 33).

Sometimes it is easier to reduce not by 11, but by another divisor factor.

Example 2. Divide 6230 by 55.

Let’s reduce the dividend and the divisor by 5 (for the dividend, we’ll discard the zero and multiply by 2): 6230: 55 = 1246: 11.
Divide 1246 by 11 “in a column”, we get 113 and 3/11.
Answer: 113 and 15/55 (the remainder is adjusted to the original divisor of 55).

B.5. Division by numbers ending in 1.

: =

Numbers ending in 1 are usually easiest to divide into columns.

B.6. Divide by numbers ending in 5.

: =

In this case, you can use the rounding method from Example B.3, long division, or the reduction by 5 method, as described here.

Example. Dividing 8117 by 65:

If the dividend is four digits, first determine the number of hundreds in the answer. To do this, divide the first pair of digits of the dividend by the divisor. Then work with the remainder of this division and the second pair. In this case: the number of hundreds is 1, the new dividend is 1617.
Round the dividend down to the tens and reduce it by 5, that is, divide by 10 and multiply 2: 1610: 5 = 161 × 2 = 322.
Divide the result by the divisor, also reduced by 5: 322: 13 = 24 and the remainder is 10.
Determine the remainder: 7 + 10 × 5 = 57. Thus, 8117: 65 = 124 and 57/65.

Multiply the hundreds of the dividend by 4: 32 × 4 = 128.
Divide the last two digits of the dividend by 25 and calculate the remainder: 68: 25 = 2 and 18 remainder.
Add the two answers: 3268: 25 = 130 and 18/25 (i.e. 130.72).

If the divisor is 75, then divide first by 25, then by 3.

B.7. Dividing three-digit numbers.

: =

First of all, determine and remember the number of tens in the answer - this will avoid a major mistake. To do this, divide the first two digits of the dividend by the divisor. For example, when dividing 943 by 34, the number of tens in the answer is 2, and when dividing 325 by 43, the number of tens is 0 (32 is less than 43).

B.8. Dividing four-digit numbers.

: =

First of all, determine and remember the number of hundreds in the answer - this will avoid a major mistake. To do this, divide the first two digits of the dividend by the divisor.
Try to apply the methods from exercises B.1-B.6, and if they do not work, divide in the usual way, “in a column.”
If the divisor is a multiple of a small number, try reducing the dividend and divisor by it. At the same time, if the dividend is not divisible by this number, discard the required number of units from it so that it is divisible (then take them into account when calculating the remainder). For a two-digit number, it is not difficult to determine whether it is factorizable - to do this, you need to check for divisibility by the numbers 2, 3, 5 and 7.

Schoolchildren learn column division, or, more correctly, the written method of dividing by a corner, already in the third grade of elementary school, but often so little attention is paid to this topic that by the 9th-11th grade not all students can use it fluently.

Division by a column by a two-digit number is taught in the 4th grade, as is division by a three-digit number, and then this technique is used only as an auxiliary technique when solving any equations or finding the value of an expression.

Obviously, by paying more attention to long division than is included in the school curriculum, the child will make it easier for him to complete math assignments up to the 11th grade. And for this you need little - to understand the topic and study, solve, keeping the algorithm in your head, to bring the calculation skill to automatism.

First, let’s briefly repeat how to divide by a column by a single-digit number:

Algorithm for dividing by a two-digit number

As with division by a single-digit number, we will sequentially move from dividing larger counting units to dividing smaller units.

1. Find the first incomplete dividend. This is a number that is divided by a divisor to produce a number greater than or equal to 1. This means that the first partial dividend is always greater than the divisor. When dividing by a two-digit number, the first partial dividend must have at least 2 digits.

Examples 76 8:24. First incomplete dividend 76
265 :53 26 is less than 53, which means it is not suitable. You need to add the next number (5). The first incomplete dividend is 265.

2. Determine the number of digits in the quotient. To determine the number of digits in a quotient, you should remember that the incomplete dividend corresponds to one digit of the quotient, and all other digits of the dividend correspond to one more digit of the quotient.

Examples 768:24. The first incomplete dividend is 76. It corresponds to 1 digit of the quotient. After the first partial divisor there is one more digit. This means that the quotient will only have 2 digits.
265:53. The first incomplete dividend is 265. It will give 1 digit of the quotient. There are no more digits in the dividend. This means that the quotient will only have 1 digit.
15344:56. The first partial dividend is 153, and after it there are 2 more digits. This means that the quotient will only have 3 digits.

3. Find the numbers in each digit of the quotient. First, let's find the first digit of the quotient. We select an integer such that when multiplied by our divisor we get a number that is as close as possible to the first incomplete dividend. We write the quotient number under the corner, and subtract the value of the product in a column from the partial divisor. We write down the remainder. We check that it is less than the divisor.

Then we find the second digit of the quotient. We rewrite the number following the first partial divisor in the dividend into the line with the remainder. The resulting incomplete dividend is again divided by the divisor and so we find each subsequent number of the quotient until the digits of the divisor run out.

4. Find the remainder(if there is).

If the digits of the quotient run out and the remainder is 0, then the division is performed without a remainder. Otherwise, the quotient value is written with a remainder.

Division by any multi-digit number (three-digit, four-digit, etc.) is also performed.

Analysis of examples of dividing by a column by a two-digit number

First, let's look at simple cases of division, when the quotient results in a single-digit number.

Let's find the value of the quotient numbers 265 and 53.

The first incomplete dividend is 265. There are no more digits in the dividend. This means that the quotient will be a single digit number.

To make it easier to choose the quotient number, let's divide 265 not by 53, but by a close round number 50. To do this, divide 265 by 10, the result will be 26 (the remainder is 5). And divide 26 by 5, there will be 5 (remainder 1). The number 5 cannot be immediately written down in the quotient, since it is a trial number. First you need to check if it fits. Let's multiply 53*5=265. We see that the number 5 has come up. And now we can write it down in a private corner. 265-265=0. The division is completed without remainder.

The quotient of 265 and 53 is 5.

Sometimes when dividing, the test digit of the quotient does not fit, and then it needs to be changed.

Let's find the value of the quotient numbers 184 and 23.

The quotient will be a single digit number.

To make it easier to choose the quotient number, let's divide 184 not by 23, but by 20. To do this, divide 184 by 10, the result will be 18 (remainder 4). And we divide 18 by 2, the result is 9. 9 is a test number, we won’t immediately write it in the quotient, but we’ll check if it’s suitable. Let's multiply 23*9=207. 207 is greater than 184. We see that the number 9 is not suitable. The quotient will be less than 9. Let's try to see if the number 8 is suitable. Let's multiply 23*8=184. We see that the number 8 is suitable. We can write it down privately. 184-184=0. The division is completed without remainder.

The quotient of 184 and 23 is 8.

Let's consider more complex cases of division.

Let's find the value of the quotient of 768 and 24.

The first incomplete dividend is 76 tens. This means that the quotient will have 2 digits.

Let's determine the first digit of the quotient. Let's divide 76 by 24. To make it easier to choose the quotient number, let's divide 76 not by 24, but by 20. That is, you need to divide 76 by 10, there will be 7 (the remainder is 6). And divide 7 by 2, you get 3 (remainder 1). 3 is the test digit of the quotient. First let's check if it fits. Let's multiply 24*3=72. 76-72=4. The remainder is less than the divisor. This means that the number 3 is suitable and now we can write it in place of the tens of the quotient. We write 72 under the first incomplete dividend, put a minus sign between them, and write the remainder under the line.

Let's continue the division. Let's rewrite the number 8 following the first incomplete dividend into the line with the remainder. We get the following incomplete dividend – 48 units. Let's divide 48 by 24. To make it easier to choose the quotient, let's divide 48 not by 24, but by 20. That is, if we divide 48 by 10, there will be 4 (the remainder is 8). And we divide 4 by 2, it becomes 2. This is the test digit of the quotient. We must first check if it will fit. Let's multiply 24*2=48. We see that the number 2 fits and, therefore, we can write it in place of the units of the quotient. 48-48=0, division is performed without remainder.

The quotient of 768 and 24 is 32.

Let's find the value of the quotient numbers 15344 and 56.

The first incomplete dividend is 153 hundreds, which means that the quotient will have three digits.

Let's determine the first digit of the quotient. Let's divide 153 by 56. To make it easier to find the quotient, let's divide 153 not by 56, but by 50. To do this, divide 153 by 10, the result will be 15 (remainder 3). And we divide 15 by 5, it becomes 3. 3 is the test digit of the quotient. Remember: you cannot immediately write it down in private, but you must first check whether it is suitable. Let's multiply 56*3=168. 168 is greater than 153. This means that the quotient will be less than 3. Let’s check if the number 2 is suitable. Multiply 56*2=112. 153-112=41. The remainder is less than the divisor, which means that the number 2 is suitable, it can be written in the place of hundreds in the quotient.

Let us form the following incomplete dividend. 153-112=41. We rewrite the number 4 following the first incomplete dividend into the same line. We get the second incomplete dividend of 414 tens. Let's divide 414 by 56. To make it more convenient to choose the quotient number, let's divide 414 not by 56, but by 50. 414:10=41(rest.4). 41:5=8(rest.1). Remember: 8 is a test number. Let's check it out. 56*8=448. 448 is greater than 414, which means that the quotient will be less than 8. Let's check if the number 7 is suitable. Multiply 56 by 7, we get 392. 414-392=22. The remainder is less than the divisor. This means that the number fits and in the quotient we can write 7 in place of tens.

We write 4 units in the line with the new remainder. This means the next incomplete dividend is 224 units. Let's continue the division. Divide 224 by 56. To make it easier to find the quotient number, divide 224 by 50. That is, first by 10, there will be 22 (the remainder is 4). And divide 22 by 5, there will be 4 (remainder 2). 4 is a test number, let's check it to see if it fits. 56*4=224. And we see that the number has come up. Let's write 4 in place of units in the quotient. 224-224=0, division is performed without remainder.

The quotient of 15344 and 56 is 274.

Example for division with remainder

To make an analogy, let's take an example similar to the example above, differing only in the last digit

Let's find the value of the quotient 15345:56

We first divide in the same way as in the example 15344:56, until we reach the last incomplete dividend 225. Divide 225 by 56. To make it easier to choose the quotient number, divide 225 by 50. That is, first by 10, there will be 22 (the remainder is 5 ). And divide 22 by 5, there will be 4 (remainder 2). 4 is a test number, let's check it to see if it fits. 56*4=224. And we see that the number has come up. Let's write 4 in place of units in the quotient. 225-224=1, division done with remainder.

The quotient of 15345 and 56 is 274 (remainder 1).

Division with zero in quotient

Sometimes in a quotient one of the numbers turns out to be 0, and children often miss it, hence the wrong solution. Let's look at where 0 can come from and how not to forget it.

Let's find the value of the quotient 2870:14

The first incomplete dividend is 28 hundreds. This means that the quotient will have 3 digits. Place three dots under the corner. This is an important point. If a child loses a zero, there will be an extra dot left, which will make them think that a number is missing somewhere.

Let's determine the first digit of the quotient. Let's divide 28 by 14. By selection we get 2. Let's check if the number 2 fits. Multiply 14*2=28. The number 2 is suitable; it can be written in place of hundreds in the quotient. 28-28=0.

The result was a zero remainder. We've marked it in pink for clarity, but you don't need to write it down. We rewrite the number 7 from the dividend into the line with the remainder. But 7 is not divisible by 14 to obtain an integer, so we write 0 in the place of tens in the quotient.

Now we rewrite the last digit of the dividend (number of units) into the same line.

70:14=5 We write the number 5 instead of the last point in the quotient. 70-70=0. There is no remainder.

The quotient of 2870 and 14 is 205.

Division must be checked by multiplication.

Division examples for self-test

Find the first incomplete dividend and determine the number of digits in the quotient.

3432:66 2450:98 15145:65 18354:42 17323:17

You have mastered the topic, now practice solving several examples in a column yourself.

1428: 42 30296: 56 254415: 35 16514: 718

Column division is an integral part of the educational material for primary school students. Further success in mathematics will depend on how correctly he learns to perform this action.

How to properly prepare a child to perceive new material?

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

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

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

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

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

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

A clear example for students and parents

Column division can be clearly explained using this example.

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

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