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 using 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.
Let's look at a simple example:
15:5=3
In this example we divided the natural number 15 completely by 3, without remainder.
Sometimes a natural number cannot be completely divided. For example, consider the problem:
There were 16 toys in the closet. There were five children in the group. Each child took the same number of toys. How many toys does each child have?
Solution:
Divide the number 16 by 5 using a column and we get:
We know that 16 cannot be divided by 5. The nearest smaller number that is divisible by 5 is 15 with a remainder of 1. We can write the number 15 as 5⋅3. As a result (16 – dividend, 5 – divisor, 3 – incomplete quotient, 1 – remainder). Got formula division with remainder which can be done checking the solution.
a=
b⋅
c+
d
a – divisible,
b - divider,
c – incomplete quotient,
d - remainder.
Answer: each child will take 3 toys and one toy will remain.
Remainder of the division
The remainder must always be less than the divisor.
If during division the remainder is zero, then this means that the dividend is divided completely or without a remainder on the divisor.
If during division the remainder is greater than the divisor, this means that the number found is not the largest. There is a greater number that will divide the dividend and the remainder will be less than the divisor.
Questions on the topic “Division with remainder”:
Can the remainder be greater than the divisor?
Answer: no.
Can the remainder be equal to the divisor?
Answer: no.
How to find the dividend using the incomplete quotient, divisor and remainder?
Answer: We substitute the values of the partial quotient, divisor and remainder into the formula and find the dividend. Formula:
a=b⋅c+d
Example #1:
Perform division with remainder and check: a) 258:7 b) 1873:8
Solution:
a) Divide by column: 
258 – dividend,
7 – divider,
36 – incomplete quotient,
6 – remainder. The remainder is less than the divisor 6<7.
7⋅36+6=252+6=258
b) Divide by column: 
1873 – divisible,
8 – divisor,
234 – incomplete quotient,
1 – remainder. The remainder is less than divisor 1<8.
Let’s substitute it into the formula and check whether we solved the example correctly:
8⋅234+1=1872+1=1873
Example #2:
What remainders are obtained when dividing natural numbers: a) 3 b)8?
Answer:
a) The remainder is less than the divisor, therefore less than 3. In our case, the remainder can be 0, 1 or 2.
b) The remainder is less than the divisor, therefore less than 8. In our case, the remainder can be 0, 1, 2, 3, 4, 5, 6 or 7.
Example #3:
What is the largest remainder that can be obtained when dividing natural numbers: a) 9 b) 15?
Answer:
a) The remainder is less than the divisor, therefore less than 9. But we need to indicate the largest remainder. That is, the number closest to the divisor. This is the number 8.
b) The remainder is less than the divisor, therefore, less than 15. But we need to indicate the largest remainder. That is, the number closest to the divisor. This number is 14.
Example #4:
Find the dividend: a) a:6=3(rest.4) b) c:24=4(rest.11)
Solution:
a) Solve using the formula:
a=b⋅c+d
(a – dividend, b – divisor, c – partial quotient, d – remainder.)
a:6=3(rest.4)
(a – dividend, 6 – divisor, 3 – partial quotient, 4 – remainder.) Let’s substitute the numbers into the formula:
a=6⋅3+4=22
Answer: a=22
b) Solve using the formula:
a=b⋅c+d
(a – dividend, b – divisor, c – partial quotient, d – remainder.)
s:24=4(rest.11)
(c – dividend, 24 – divisor, 4 – partial quotient, 11 – remainder.) Let’s substitute the numbers into the formula:
с=24⋅4+11=107
Answer: c=107
Task:
Wire 4m. need to be cut into 13cm pieces. How many such pieces will there be?
Solution:
First you need to convert meters to centimeters.
4m.=400cm.
We can divide by a column or in our mind we get:
400:13=30(remaining 10)
Let's check:
13⋅30+10=390+10=400
Answer: You will get 30 pieces and 10 cm of wire will remain.
How to teach a child division? The simplest method is learn long division. This is much easier than carrying out calculations in your head; it helps you avoid getting confused, not “losing” the numbers, and developing a mental scheme that will work automatically in the future.
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How is it carried out?
Division with a remainder is a method in which a number cannot be divided into exactly several parts. As a result of this mathematical operation, in addition to the whole part, an indivisible piece remains.
Let's give a simple example how to divide with remainder:
There is a jar for 5 liters of water and 2 jars of 2 liters each. When water is poured from a five-liter jar into two-liter jars, 1 liter of unused water will remain in the five-liter jar. This is the remainder. In digital form it looks like this:
5:2=2 rest (1). Where is 1 from? 2x2=4, 5-4=1.
Now let's look at the order of division into a column with a remainder. This visually simplifies the calculation process and helps not to lose numbers.
The algorithm determines the location of all elements and the sequence of actions by which the calculation is performed. As an example, let's divide 17 by 5.
Main stages:
- Correct entry. Dividend (17) – located on the left side. To the right of the dividend, write the divisor (5). A vertical line is drawn between them (indicating the division sign), and then, from this line, a horizontal line is drawn, emphasizing the divisor. The main features are indicated in orange.
- Search for the whole. Next, the first and simplest calculation is carried out - how many divisors fit into the dividend. Let's use the multiplication table and check in order: 5*1=5 - fits, 5*2=10 - fits, 5*3=15 - fits, 5*4=20 - doesn't fit. Five times four is more than seventeen, which means the fourth five does not fit. Let's go back to three. A 17 liter jar will fit 3 five liter jars. We write the result in the form: 3 is written under the line, under the divisor. 3 is an incomplete quotient.
- Definition of remainder. 3*5=15. We write 15 under the dividend. We draw a line (indicated by the “=” sign). Subtract the resulting number from the dividend: 17-15=2. We write the result below the line - in a column (hence the name of the algorithm). 2 is the remainder.
Note! When dividing in this way, the remainder must always be less than the divisor.
When the divisor is greater than the dividend
Difficulty arises when the divisor is larger than the dividend. Decimal fractions are not yet studied in the 3rd grade curriculum, but following logic, the answer should be written as a fraction - at best a decimal, at worst a simple one. But (!) in addition to the program, the calculation method limited by the task: it is necessary not to divide, but to find the remainder! some of them are not! How to solve such a problem?
Note! There is a rule for cases when the divisor is greater than the dividend: the partial quotient is equal to 0, the remainder is equal to the dividend.
How to divide the number 5 by the number 6, highlighting the remainder? How many 6-liter cans will fit into a 5-liter jar? because 6 is greater than 5.
The assignment requires filling 5 liters - not a single one has been filled. This means that all 5 remain. Answer: partial quotient = 0, remainder = 5.
Division begins to be studied in the third grade of school. By this time, students should already be able to do the division of two-digit numbers by single-digit numbers.
Solve the problem: 18 sweets need to be distributed to five children. How many candies will be left?
Examples:
We find the incomplete quotient: 3*1=3, 3*2=6, 3*3=9, 3*4=12, 3*5=15. 5 – overkill. Let's go back to 4.
Remainder: 3*4=12, 14-12=2.
Answer: incomplete quotient 4, 2 left.
You may ask why when divided by 2, the remainder is either 1 or 0. According to the multiplication table, between digits that are multiples of two there is a difference of one.
Another task: 3 pies must be divided into two.
Divide 4 pies between two.
Divide 5 pies between two.
Working with multi-digit numbers
The 4th grade program offers a more complex process of division with increasing calculated numbers. If in the third grade calculations were carried out on the basis of a basic multiplication table ranging from 1 to 10, then fourth graders carry out calculations with multi-digit numbers over 100.
It is most convenient to perform this action in a column, since the incomplete quotient will also be a two-digit number (in most cases), and the column algorithm simplifies the calculations and makes them more visual.
Let's divide multi-digit numbers to double digits: 386:25
This example differs from the previous ones in the number of calculation levels, although the calculations are carried out according to the same principle as before. Let's take a closer look:
386 is the dividend, 25 is the divisor. It is necessary to find the incomplete quotient and select the remainder.
First level
The divisor is a two-digit number. The dividend is three-digit. We select the first two left digits of the dividend - this is 38. We compare them with the divisor. Is 38 more than 25? Yes, that means 38 can be divided by 25. How many whole 25 are in 38?
25*1=25, 25*2=50. 50 is more than 38, let's go back one step.
Answer - 1. Write the unit to the zone not completely private.
38-25=13. Write the number 13 below the line.
Second level
Is 13 more than 25? No - that means you can “lower” the number 6 down by adding it next to 13, on the right. It turned out to be 136. Is 136 more than 25? Yes - that means you can subtract it. How many times can 25 fit into 136?
25*1=25, 25*2=50, 25*3=75, 25*4=100, 25*5=125, 256*=150. 150 is more than 136 – we go back one step. We write the number 5 in the incomplete quotient zone, to the right of one.
Calculate the remainder:
136-125=11. Write it below the line. Is 11 more than 25? No - division cannot be carried out. Does the dividend have digits left? No - there is nothing more to share. The calculations are completed.
Answer: the partial quotient is 15, the remainder is 11.
What if such a division is proposed, when the two-digit divisor is greater than the first two digits of the multi-digit dividend? In this case, the third (fourth, fifth and subsequent) digit of the dividend takes part in the calculations immediately.
Let's give examples for division with three- and four-digit numbers:
75 is a two-digit number. 386 – three-digit. Compare the first two digits on the left with the divisor. 38 is more than 75? No - division cannot be carried out. We take all 3 numbers. Is 386 more than 75? Yes, division can be done. We carry out calculations.
75*1=75, 75*2=150, 75*3=225, 75*4=300, 75*5= 375, 75*6=450. 450 is more than 386 – we go back a step. We write 5 in the incomplete quotient zone.
The easiest way to divide multi-digit numbers is with a column. Column division is also called corner division.
Before we begin to perform division by a column, we will consider in detail the very form of recording division by a column. First, write down the dividend and put a vertical line to the right of it:
Behind the vertical line, opposite the dividend, write the divisor and draw a horizontal line under it:
Under the horizontal line, the resulting quotient will be written step by step:
Intermediate calculations will be written under the dividend:
The full form of writing division by column is as follows:
How to divide by column
Let's say we need to divide 780 by 12, write the action in a column and proceed to division:
Column division is performed in stages. The first thing we need to do is determine the incomplete dividend. We look at the first digit of the dividend:
this number is 7, since it is less than the divisor, we cannot start division from it, which means we need to take another digit from the dividend, the number 78 is greater than the divisor, so we start division from it:
In our case the number 78 will be incomplete divisible, it is called incomplete because it is only a part of the divisible.
Having determined the incomplete dividend, we can find out how many digits will be in the quotient, for this we need to calculate how many digits are left in the dividend after the incomplete dividend, in our case there is only one digit - 0, this means that the quotient will consist of 2 digits.
Having found out the number of digits that should be in the quotient, you can put dots in its place. If, when completing the division, the number of digits turns out to be more or less than the indicated points, then an error was made somewhere:
Let's start dividing. We need to determine how many times 12 is contained in the number 78. To do this, we sequentially multiply the divisor by the natural numbers 1, 2, 3, ... until we get a number as close as possible to the incomplete dividend or equal to it, but not exceeding it. Thus, we get the number 6, write it under the divisor, and from 78 (according to the rules of column subtraction) we subtract 72 (12 · 6 = 72). After we subtract 72 from 78, the remainder is 6:
Please note that the remainder of the division shows us whether we have chosen the number correctly. If the remainder is equal to or greater than the divisor, then we did not choose the number correctly and we need to take a larger number.
To the resulting remainder - 6, add the next digit of the dividend - 0. As a result, we get an incomplete dividend - 60. Determine how many times 12 is contained in the number 60. We get the number 5, write it in the quotient after the number 6, and subtract 60 from 60 ( 12 5 = 60). The remainder is zero:
Since there are no more digits left in the dividend, it means 780 is divided by 12 completely. As a result of performing long division, we found the quotient - it is written under the divisor:
Let's consider an example when the quotient results in zeros. Let's say we need to divide 9027 by 9.
We determine the incomplete dividend - this is the number 9. We write 1 into the quotient and subtract 9 from 9. The remainder is zero. Usually, if in intermediate calculations the remainder is zero, it is not written down:
We take down the next digit of the dividend - 0. We remember that when dividing zero by any number there will be zero. We write zero into the quotient (0: 9 = 0) and subtract 0 from 0 in intermediate calculations. Usually, in order not to clutter up intermediate calculations, calculations with zero are not written:
We take down the next digit of the dividend - 2. In intermediate calculations it turned out that the incomplete dividend (2) is less than the divisor (9). In this case, write zero to the quotient and remove the next digit of the dividend:
We determine how many times 9 is contained in the number 27. We get the number 3, write it as a quotient, and subtract 27 from 27. The remainder is zero:
Since there are no more digits left in the dividend, it means that the number 9027 is divided by 9 completely:
Let's consider an example when the dividend ends in zeros. Let's say we need to divide 3000 by 6.
We determine the incomplete dividend - this is the number 30. We write 5 into the quotient and subtract 30 from 30. The remainder is zero. As already mentioned, it is not necessary to write zero in the remainder in intermediate calculations:
We take down the next digit of the dividend - 0. Since dividing zero by any number will result in zero, we write zero in the quotient and subtract 0 from 0 in intermediate calculations:
We take down the next digit of the dividend - 0. We write another zero into the quotient and subtract 0 from 0 in intermediate calculations. Since in intermediate calculations, calculations with zero are usually not written down, the entry can be shortened, leaving only the remainder - 0. Zero in the remainder in at the very end of the calculation is usually written to show that the division is complete:
Since there are no more digits left in the dividend, it means 3000 is divided by 6 completely:
Column division with remainder
Let's say we need to divide 1340 by 23.
We determine the incomplete dividend - this is the number 134. We write 5 into the quotient and subtract 115 from 134. The remainder is 19:
We take down the next digit of the dividend - 0. We determine how many times 23 is contained in the number 190. We get the number 8, write it into the quotient, and subtract 184 from 190. We get the remainder 6:
Since there are no more digits left in the dividend, the division is over. The result is an incomplete quotient of 58 and a remainder of 6:
1340: 23 = 58 (remainder 6)
It remains to consider an example of division with a remainder, when the dividend is less than the divisor. Let us need to divide 3 by 10. We see that 10 is never contained in the number 3, so we write 0 as a quotient and subtract 0 from 3 (10 · 0 = 0). Draw a horizontal line and write down the remainder - 3:
3: 10 = 0 (remainder 3)
Long division calculator
This calculator will help you perform long division. Simply enter the dividend and divisor and click the Calculate button.
How to divide decimals by natural numbers? Let's look at the rule and its application using examples.
To divide a decimal fraction by a natural number, you need to:
1) divide the decimal fraction by the number, ignoring the comma;
2) when the division of the whole part is completed, put a comma in the quotient.
Examples.
Divide decimals:
To divide a decimal fraction by a natural number, divide without paying attention to the comma. 5 is not divisible by 6, so we put zero in the quotient. The division of the whole part is completed, we put a comma in the quotient. We take down the zero. Divide 50 by 6. Take 8. 6∙8=48. From 50 we subtract 48, the remainder is 2. We take away 4. We divide 24 by 6. We get 4. The remainder is zero, which means the division is over: 5.04: 6 = 0.84.
2) 19,26: 18
Divide the decimal fraction by a natural number, ignoring the comma. Divide 19 by 18. Take 1 each. The division of the whole part is completed, put a comma in the quotient. We subtract 18 from 19. The remainder is 1. We take away 2. 12 is not divisible by 18, and in the quotient we write zero. We take down 6. We divide 126 by 18, we get 7. The division is over: 19.26: 18 = 1.07.
Divide 86 by 25. Take 3 each. 25∙3=75. From 86 we subtract 75. The remainder is 11. The division of the whole part is completed, in the quotient we put a comma. We take down 5. We take 4 each. 25∙4=100. From 115 we subtract 100. The remainder is 15. We remove zero. We divide 150 by 25. We get 6. The division is over: 86.5: 25 = 3.46.
4) 0,1547: 17
Zero is not divisible by 17; we write zero in the quotient. The division of the whole part is completed, we put a comma in the quotient. We take down 1. 1 is not divisible by 17, we write zero in the quotient. We take down 5. 15 is not divisible by 17, we write zero in the quotient. We take down 4. We divide 154 by 17. We take 9 each. 17∙9=153. From 154 we subtract 153. The remainder is 1. We take away 7. We divide 17 by 17. We get 1. The division is over: 0.1547: 17 = 0.0091.
5) A decimal fraction can also be obtained when dividing two natural numbers.
When dividing 17 by 4, we take 4 each. The division of the whole part is completed, in the quotient we put a comma. 4∙4=16. From 17 we subtract 16. The remainder is 1. We remove zero. Divide 10 by 4. Take 2. 4∙2=8. From 10 we subtract 8. The remainder is 2. We remove zero. Divide 20 by 4. Take 5 each. Division is completed: 17: 4 = 4.25.
And a couple more examples of dividing decimals by natural numbers:
