How to divide 4-digit numbers. Division of natural numbers by a column, examples, solutions

Tasks on the topic: "Division. Division of multi-digit numbers by a column"

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Teaching aids and simulators in the Integral online store for grade 4
Manual for the textbook M.I. Moro Manual for the textbook L.G. Peterson

Dividing two-digit numbers by one-digit number

1. Write down the given sentences in the form of numerical expressions and solve them.

1.1. Divide 72 by 8.

1.2. Divide 81 by 9.

1.3. Divide 62 by 21.

2. Perform the division of the numbers.

Solving word problems for dividing a multi-digit number by a single-digit number

1. How many notebooks of 14 rubles can be bought for 84 rubles?

2. The apple harvest was 81 kg. How many boxes do you need to arrange apples if one box holds 9 kg?

3. The car transports 7 tons of sand for 1 trip. How many trips does he need to make to transport 140 tons of sand?

4. It is necessary to transport 176 kg of sugar from the warehouse to the store. How many bags for transporting sugar are required if the bag holds 8 kg of sugar?

5. One square meter of flooring requires 14 kg of cement. How many square meters is 126 kg of cement sufficient for?

Dividing a multi-digit number by a two-digit number

1. Perform division.

Solving word problems for dividing a multi-digit number by a multi-digit number

1. The farmer has harvested cabbage and onions. He harvested 10 455 kg of cabbage, and 123 times less onion. How many kg of onions did the farmer collect?

2. Three guys divided the number 26668 by 59. The first got 457, the second - 452, and the third - 251. Which answer is correct?

3. For the winter, the farmer prepared 2,720 kg of mixed feed for sheep. For each sheep, 85 kg are harvested. How many sheep does the farmer have?

4. In the school garden, 13 beds of equal length carrots were planted. A total of 5863 kg of carrots were harvested. How many kg of carrots were collected from each garden bed?

Division it is convenient to write multidigit or multidigit numbers in a column... Let's see how to do this. Let's start by dividing a multi-bit number by a one-bit number, and gradually increase the bit width of the dividend.

So let's divide 354 on the 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 write the quotient under the divisor.

Now we start dividing 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 divider.

3 more 2 , means 3 and there is an incomplete dividend. We put a full stop in the quotient and determine how many more digits there will be in the quotient - the same number left in the dividend after highlighting the incomplete dividend. In our case, there are as many digits in the quotient as in the dividend, that is, the most significant digit will be hundreds:

To 3 split into 2 remember the multiplication table by 2 and find the number when multiplied by 2, we get the largest product, which is less than 3.

2 × 1 = 2 (2< 3)

2 × 2 = 4 (4> 3)

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

We write down 1 in the quotient in place of the first point (in the category of hundreds), and write the found product under the dividend:

Now we find the difference between the first incomplete dividend and the product of the found digit of the 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 the 2 again we recall the multiplication table 2 and find the largest product that is less 15 :

2 × 7 = 14 (14< 15)

2 × 8 = 16 (16> 15)

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

We continue to divide, for which we find third incomplete dividend... We lower the next bit of the dividend:

We divide the incomplete divisible by 2, we put 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, we find the difference:

We got the difference equal to zero, which means the division has been made right.

Let's complicate the task and give another example:

1020 ÷ 5

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

Thousands of the dividend is 1 , compare with the divisor:

1 < 5

Add hundreds to the incomplete dividend and compare:

10 > 5 - we found an incomplete dividend.

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

10 – 10 = 0

0 we do not write, we omit the next place of the dividend - the place of tens:

Compare the second incomplete dividend with the divisor.

2 < 5

We should add one more digit to the incomplete dividend, for this in the quotient, we put 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 long division:

1. If there are zeros in the dividend and the divisor in the lower digits, then they can be canceled before dividing, for example:

How many zeros in the least significant bit of the dividend we remove, the same number of zeros in the least significant bits of the divisor.

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

So, let's formulate the sequence of actions for long division.

1. Place the dividend on the left, the divisor on the right. Remember that we divide the dividend by separating the 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 there are zeros in the dividend and the divisor in the lower digits, then they can be canceled before the division.
3. Determine the first incomplete divisor:

but) select the highest bit of the dividend into an incomplete divisor;

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

in) add the next digit to the incomplete dividend and go to paragraph (b).

1. We determine how many digits there will be in the quotient, and put as many points in place of the quotient (under the divisor) how many digits there will be 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 the allocation of the incomplete dividend.
2. We divide the incomplete dividend by the divisor, for this we find the number, when multiplied by the divisor, the number would be either equal to the incomplete dividend, or less than it.
3. The found number is written in place of the next digit of the quotient (dot), and the result of multiplying it by the divisor is written under the incomplete dividend and we find their difference.
4. If the found difference 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 in the dividend, then we continue the division, otherwise go to step 10 .
6. We lower the next digit of the dividend to the difference and get the next incomplete dividend:

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

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

c) go to point (a).

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

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

One of the important stages in teaching a child to do math is learning how to divide prime numbers. How to explain the division to a child, when can you start mastering this topic?

In order to teach a child to divide, it is necessary that by the time of learning he has already mastered such mathematical operations as addition, subtraction, and also have a clear idea of ​​the very essence of the actions 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 already wrote about this article may be useful for you.

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

At this stage, it is necessary to form the child's understanding that division is the division of something into equal parts. The easiest way to teach a child to do this is to invite him to share a number of objects between his friends or family members.

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

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 in the next step, when the child needs to understand that division is the inverse 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, using any example, the relationship between multiplication and division.

Example: 4x2 = 8. Remind your child that the product of multiplication is the product of two numbers. Then 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 another factor that was not used in the operation.

You also need to teach the young student how the categories describing the division operation are called - "dividend", "divisor" and "quotient". Using an example, show which numbers are divisible, divisor, and quotient. Reinforce this knowledge, they are necessary for further learning!

In fact, you need to teach your child the multiplication table "vice versa", and you need to remember it as well as the multiplication table itself, because this will be necessary when you start learning long division.

Divide by column - give an example

Before starting the lesson, remember with your child what the numbers are called during the division operation. What is "divisor", "divisible", "quotient"? Teach you 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 and 7 is the divisor. The result will be the quotient, which is what you need to calculate.

Step 1... We write down the numbers, dividing them with a "corner".

Step 2. Show the student the number of the dividend and ask him to choose the smallest number that is greater than the divisor. Of the three digits 9, 3 and 8, this number is 9. Ask 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. We proceed to the design of the division by a column:

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

We write down the result.

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

Step 5. Next, we act according to the already known algorithm. We 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 down below under the number 23 in a column.

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

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

Step 7 We analyze how many times our divisor 7 is contained in the resulting number? That's right, 4 times. We write the resulting figure into the result. So, we got the quotient obtained as a result of division by a long bar = 134.

How to teach a child to divide - consolidate the skill

The main reason why many schoolchildren have a problem with mathematics is the inability to quickly do simple arithmetic calculations. And on this basis, all mathematics in elementary school is built. 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 the mind, a correct teaching methodology and skill consolidation are needed. To do this, we advise you to use the currently popular tutorials in mastering the division skill. Some are designed for children to study with their parents, others for independent work.

1. "Division. Level 3. Workbook "from the largest international center for continuing education Kumon
2. "Division. Level 4. Workbook "by Kumon
3. “Not Mental arithmetic. A system for teaching a child to multiply and divide quickly. For 21 days. Notebook simulator. " from Sh. Akhmadulin - the author of educational bestselling books

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

If the child is good at using the multiplication table and "reverse" division, he will not have any difficulties. Nevertheless, it is very important to constantly train the acquired skill. Do not stop there once you understand that the child has grasped the essence of the method.

In order to easily teach a child the division operation, you need:

• So that at the age of two or three years, he mastered the relationship "whole - part". He should develop an understanding of the whole as an indivisible 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 at primary school age the child can freely operate with actions of adding and subtracting numbers, understand the essence of the processes of multiplication and division.

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

Therefore, encourage and develop the child's observation skills, draw analogies with mathematical actions (operations on counting and division, analysis of the relationship "part-whole", etc.) during construction, games and observations of nature.

Teacher, specialist of the children's development center
Druzhinina Elena
site specially for the project

Video plot for parents, how to correctly explain long division to a child:

Children in grade 2-3 master a new mathematical action - division. It is not easy for a student to grasp the essence of this mathematical action, 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 with 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 a fun game.

You can set tasks in this way:

1 Provide a play-based learning space for your child. Place his toys in a circle, and give the child pears or candy. Have a student divide 4 candies between 2 or 3 dolls. To gain understanding on the part of the child, gradually add the number of candies to 8 and 10. Even if the baby will act for a long time, do not press or shout at him. You will need patience. If the child does something wrong, correct it calmly. Then, as he completes the first action of dividing the candies between the participants in the game, he will ask him to calculate how many candies each toy got. Now the conclusion. If there were 8 candies and 4 toys, then each got 2 candies. Let your child know that sharing means distributing an equal amount of candy to all toys.

2 You can teach mathematical action using numbers. Let the student know that numbers qualify as pears or candy. Tell them that the number of pears you want to divide is the dividend. And the number of toys containing sweets is a divisor.

3 Give your child 6 pears. Challenge him to divide the number of pears between grandfather, dog, and dad. Then ask him to divide 6 pears between grandfather and dad. Explain to your child the reason why the division is not the same.

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

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

Don't overload your child with new knowledge. You need to teach gradually. You need to move on to a new material when the previous material is fixed.

Learning long division using the multiplication table

Pupils up to grade 5 will be able to figure out division faster, provided that they know multiplication well.

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

• Tell the student to arbitrarily multiply the values ​​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. As a result of the mathematical operation, you get 5. The student will be able to make sure that division is the same as multiplication, but vice versa.

You can use the multiplication table for clarity of division, if the child has mastered it well.

Learning long division in a notebook

You need to start learning when the student understands the material about division in practice, using the game and the multiplication table.

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

Explain the mathematical operation in detail:

• Write an example in your notebook: 105 divided by 5.
• Write it down like long division.
• Tell us that 105 is the dividend and 5 is the divisor.
• With the student, identify 1 digit that allows division. The value of the dividend is 1, this figure is not divisible by 5. But the second number is 0. As a result, you get 10, this value is allowed to divide this example. The number 5 is twice included in the number 10.
• In the division column, under the number 5, write the number 2.
• Ask the child to multiply the number 5 by 2. The result of the multiplication will be 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 obtained as a result of the subtraction - 0. 105 has a number left that did not participate in the division - 5. This number must be written down.
• As a result, you get 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 after k 232, 342, 345 , etc.

Learning to divide with remainder

When the child has mastered the material about division, the task can be complicated. Division with remainder is the next step in learning. You need to explain using the available examples:

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

Simple examples for a child

Using the same example, you can continue:

• When dividing 35 by 8, the remainder is 3. Add 0 to the remainder. In this case, after the number 4 in the column, you need to put a comma. The result will now be fractional.
• When dividing 30 by 8, you get 3. This figure must be written after the decimal point.
• Now you need to write 24 under the value 30 (the result of multiplying 8 by 3). As a result, you get 6. Zero also needs to be added to the number 6. It turns out 60.
• Number 8 is placed in the number 8 is included 7 times. That is, you get 56.
• If you subtract 60 from 56, you get 4. This number also needs to be signed 0. It turns out 40. In the multiplication table, the child can see that 40 is the result of multiplying 8 by 5. That is, the number 8 is included in the number 40 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 the values ​​many times, which will have a remainder.

Learning division through games

Parents can use division games to teach students. You can give your child coloring pages in which you need to determine the color of the pencil by dividing. It is necessary 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, which will contain the results of the division. And the colors to use are examples. For example, red is marked with an example: 15 divided by 3. It turns out 5. You need to find a part of the picture under this number and color it. Math coloring is fun for kids. Therefore, parents should try this teaching method.

Learning to divide the smallest number by the largest number

This division assumes that the quotient starts at 0, 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.

Dividing multidigit numbers is easiest to do with a column. Division by a column is also called division by corner.

Before starting to perform long division, consider in detail the very form of recording long division. First, write the dividend and put a vertical bar to the right of it:

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

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

Intermediate calculations will be written under the dividend:

The full form of writing column division is as follows:

How to divide by a column

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

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

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

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

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

Having learned the number of digits that should turn out in the quotient, 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 an error was made somewhere:

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

Note that the remainder of the division tells us if we have chosen the correct number. If the remainder is equal to or greater than the divisor, then we have chosen the wrong 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 get an incomplete dividend - 60. Determine how many times 12 is contained in the number 60. We get the number 5, write it in the quotient after the number 6, and subtract 60 from 60 ( 12 5 = 60). The remainder is zero:

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

Consider an example when the quotient is zeros. Let's say we need to divide 9027 by 9.

Determine the incomplete dividend - this is the number 9. Write in the quotient 1 and subtract 9. The remainder is zero. Usually, if in intermediate calculations the remainder turns out to be zero, it is not written:

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

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 in the quotient and the next digit of the dividend is demolished:

Determine how many times 9 is contained in the number 27. We get the number 3, write it down in the 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 was divided by 9 completely:

Consider an example where the dividend is zero-terminated. Let's say we need to divide 3000 by 6.

Determine the incomplete dividend - this is the number 30. We write in the quotient 5 and subtract 30 from 30. The remainder is zero. As already mentioned, it is not necessary to write a 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 down to the quotient zero and in intermediate calculations we subtract 0 from 0:

We demolish the next digit of the dividend - 0. We write another zero into the quotient and subtract 0 from 0 in intermediate calculations. Since in intermediate calculations, calculations with zero are usually not written, the record can be shortened, leaving only the remainder - 0. Zero in the remainder in the very end of the calculation is usually written in order to show that the division is performed entirely:

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

Column division with remainder

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

Determine the incomplete dividend - this is the number 134. We write in the quotient 5 and subtract 115 from 134. The remainder is 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 down in the quotient, and subtract 184 from 190. We get the remainder 6:

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

1340: 23 = 58 (remainder 6)

It remains to consider an example of division with 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 0 in the quotient and subtract 0 from 3 (10 · 0 = 0). We draw a horizontal line and write down the remainder - 3:

3: 10 = 0 (remainder 3)

Long division calculator

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