Division is one of the four basic mathematical operations (addition, subtraction, multiplication). Division, like other operations, is important not only in mathematics, but also in Everyday life. For example, you as a whole class (25 people) donate money and buy a gift for the teacher, but you don’t spend it all, there will be change left over. So you will need to divide the change among everyone. The division operation comes into play to help you solve this problem.
Division is an interesting operation, as we will see in this article!
Dividing numbers
So, a little theory, and then practice! What is division? Division is breaking something into equal parts. That is, it could be a bag of sweets that needs to be divided into equal parts. For example, there are 9 candies in a bag, and the person who wants to receive them is three. Then you need to divide these 9 candies among three people.
It is written like this: 9:3, the answer will be the number 3. That is, dividing the number 9 by the number 3 shows the number of three numbers contained in the number 9. The reverse action, a check, will be multiplication. 3*3=9. Right? Absolutely.
So let's look at example 12:6. First, let's name each component of the example. 12 – dividend, that is. a number that can be divided into parts. 6 is a divisor, this is the number of parts into which the dividend is divided. And the result will be a number called “quotient”.
Let's divide 12 by 6, the answer will be the number 2. You can check the solution by multiplying: 2*6=12. It turns out that the number 6 is contained 2 times in the number 12.
Division with remainder
What is division with a remainder? This is the same division, only the result is not an even number, as shown above.
For example, let's divide 17 by 5. Since the largest number divisible by 5 to 17 is 15, then the answer will be 3 and the remainder is 2, and is written like this: 17:5 = 3(2).
For example, 22:7. In the same way, we determine the maximum number divisible by 7 to 22. This number is 21. The answer then will be: 3 and the remainder 1. And it is written: 22:7 = 3 (1).
Division by 3 and 9
A special case of division would be division by the number 3 and the number 9. If you want to find out whether a number is divisible by 3 or 9 without a remainder, then you will need:
Find the sum of the digits of the dividend.
Divide by 3 or 9 (depending on what you need).
If the answer is obtained without a remainder, then the number will be divided without a remainder.
For example, the number 18. The sum of the digits is 1+8 = 9. The sum of the digits is divisible by both 3 and 9. The number 18:9=2, 18:3=6. Divided without remainder.
For example, the number 63. The sum of the digits is 6+3 = 9. Divisible by both 9 and 3. 63:9 = 7, and 63:3 = 21. Such operations are carried out with any number to find out whether it is divisible with the remainder by 3 or 9, or not.
Multiplication and division
Multiplication and division are opposite operations. Multiplication can be used as a test for division, and division can be used as a test for multiplication. You can learn more about multiplication and master the operation in our article about multiplication. Which describes multiplication in detail and how to do it correctly. There you will also find the multiplication table and examples for training.
Here is an example of checking division and multiplication. Let's say the example is 6*4. Answer: 24. Then let's check the answer by division: 24:4=6, 24:6=4. It was decided correctly. In this case, the check is performed by dividing the answer by one of the factors.
Or an example is given for the division 56:8. Answer: 7. Then the test will be 8*7=56. Right? Yes. In this case, the test is performed by multiplying the answer by the divisor.
Division 3 class
In third grade they are just starting to go through division. Therefore, third graders solve the simplest problems:
Problem 1. A factory worker was given the task of putting 56 cakes into 8 packages. How many cakes should be put in each package to make the same amount in each?
Problem 2. On New Year's Eve at school, children in a class of 15 students were given 75 candies. How many candies should each child receive?
Problem 3. Roma, Sasha and Misha picked 27 apples from the apple tree. How many apples will each person get if they need to be divided equally?
Problem 4. Four friends bought 58 cookies. But then they realized that they could not divide them equally. How many additional cookies do the kids need to buy so that each gets 15?
Division 4th grade
The division in the fourth grade is more serious than in the third. All calculations are carried out using the column division method, and the numbers involved in the division are not small. What is long division? You can find the answer below:
Column division
What is long division? This is a method that allows you to find the answer to dividing large numbers. If prime numbers like 16 and 4 can be divided, and the answer is clear - 4. Then 512:8 is not easy for a child in his mind. And it’s our task to talk about the technique for solving such examples.
Let's look at an example, 512:8.
1 step. Let's write the dividend and divisor as follows:
The quotient will ultimately be written under the divisor, and the calculations under the dividend.
Step 2. We start dividing from left to right. First we take the number 5: 
Step 3. The number 5 is less than the number 8, which means it will not be possible to divide. Therefore, we take another digit of the dividend:
Now 51 is greater than 8. This is an incomplete quotient.
Step 4. We put a dot under the divisor.
Step 5. After 51 there is another number 2, which means there will be one more number in the answer, that is. quotient is a two-digit number. Let's put the second point:
Step 6. We begin the division operation. The largest number divisible by 8 without a remainder to 51 is 48. Dividing 48 by 8, we get 6. Write the number 6 instead of the first dot under the divisor:
Step 7. Then write the number exactly below the number 51 and put a “-” sign:
Step 8. Then we subtract 48 from 51 and get the answer 3.
* 9 step*. We take down the number 2 and write it next to the number 3:
Step 10 We divide the resulting number 32 by 8 and get the second digit of the answer – 4.
So the answer is 64, without remainder. If we divided the number 513, then the remainder would be one.
Division of three digits
Dividing three-digit numbers is done using the long division method, which was explained in the example above. An example of just a three-digit number.
Division of fractions
Dividing fractions is not as difficult as it seems at first glance. For example, (2/3):(1/4). The method of this division is quite simple. 2/3 is the dividend, 1/4 is the divisor. You can replace the division sign (:) with multiplication ( ), but to do this you need to swap the numerator and denominator of the divisor. That is, we get: (2/3)(4/1), (2/3)*4, this is equal to 8/3 or 2 integers and 2/3. Let's give another example, with an illustration for better understanding. Consider the fractions (4/7):(2/5):
As in the previous example, we reverse the 2/5 divisor and get 5/2, replacing division with multiplication. We then get (4/7)*(5/2). We make a reduction and answer: 10/7, then take out the whole part: 1 whole and 3/7.
Dividing numbers into classes
Let's imagine the number 148951784296, and divide it into three digits: 148,951,784,296. So, from right to left: 296 is the class of units, 784 is the class of thousands, 951 is the class of millions, 148 is the class of billions. In turn, in each class 3 digits have their own digit. From right to left: the first digit is units, the second digit is tens, the third is hundreds. For example, the class of units is 296, 6 is ones, 9 is tens, 2 is hundreds.
Division of natural numbers
Division of natural numbers is the simplest division described in this article. It can be either with or without a remainder. The divisor and dividend can be any non-fractional, integer numbers. 
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Division presentation
Presentation is another way to visualize the topic of division. Below we will find a link to an excellent presentation that does a good job of explaining how to divide, what division is, what dividend, divisor and quotient are. Don’t waste your time, but consolidate your knowledge!
Examples for division
Easy level
Average level
Difficult level
Games for developing mental arithmetic
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The game “Guess the Operation” develops thinking and memory. The main point game, you need to choose a mathematical sign for the equality to be true. There are examples on the screen, look carefully and put the right sign"+" or "-" so that the equality is true. The “+” and “-” signs are located at the bottom of the picture, select the desired sign and click on the desired button. If you answered correctly, you score points and continue playing.
Game "Simplification"
The game “Simplification” develops thinking and memory. The main essence of the game is to quickly perform a mathematical operation. A student is drawn on the screen at the blackboard, and a mathematical operation is given; the student needs to calculate this example and write the answer. Below are three answers, count and click the number you need using the mouse. If you answered correctly, you score points and continue playing.
Game "Quick addition"
The game "Quick Addition" develops thinking and memory. The main essence of the game is to choose numbers whose sum is equal to a given number. In this game, a matrix from one to sixteen is given. A given number is written above the matrix; you need to select the numbers in the matrix so that the sum of these digits is equal to the given number. If you answered correctly, you score points and continue playing.
Visual Geometry Game
The game "Visual Geometry" develops thinking and memory. The main essence of the game is to quickly count the number of shaded objects and select it from the list of answers. In this game, blue squares are shown on the screen for a few seconds, you need to quickly count them, then they close. Below the table there are four numbers written, you need to select one correct number and click on it with the mouse. If you answered correctly, you score points and continue playing.
Game "Piggy Bank"
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Game "Fast addition reload"
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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.
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 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. Let's 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 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
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 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. So parents should try this method training.
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.
Column? How can you independently practice the skill of long division at home if your child did not learn something at school? Dividing by columns is taught in grades 2-3; for parents, of course, this is a passed stage, but if you wish, you can remember the correct notation and explain in an understandable way to your student what he will need in life.
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What should a 2nd-3rd grade child know to learn to do long division?
How to correctly explain division to a 2-3 grade child so that he doesn’t have problems in the future? First, let's check if there are any gaps in knowledge. Make sure that:
- the child can freely perform addition and subtraction operations;
- knows the digits of numbers;
- knows by heart.
How to explain to a child the meaning of the action “division”?
- Everything needs to be explained to the child using a clear example.
Ask to share something among family members or friends. For example, candy, pieces of cake, etc. It is important that the child understands the essence - you need to divide equally, i.e. without a trace. Practice with different examples.
Let's say 2 groups of athletes must take seats on the bus. We know how many athletes are in each group and how many seats there are on the bus. You need to find out how many tickets one and the other group need to buy. Or 24 notebooks should be distributed to 12 students, as many as each gets.
- When your child understands the principle of division, show mathematical notation of this operation, name the components.
- Explain that Division is the opposite operation of multiplication, multiplication inside out.
It is convenient to show the relationship between division and multiplication using a table as an example.
For example, 3 times 4 equals 12.
3 is the first multiplier;
4 - second factor;
12 is the product (the result of multiplication).
If 12 (the product) is divided by 3 (the first factor), we get 4 (the second factor).
Components when divided are called differently:
12 - dividend;
3 - divider;
4 - quotient (result of division).
How to explain to a child the division of a two-digit number by a single-digit number not in a column?
For us adults, it’s easier to write “in the corner” the old fashioned way – and that’s the end of it. BUT! Children have not yet completed long division, what should they do? How to teach a child to divide a two-digit number by a single-digit number without using column notation?
Let's take 72:3 as an example.
It's simple! We break down 72 into numbers that can easily be divided verbally by 3:
72=30+30+12.
Everything immediately became clear: we can divide 30 by 3, and a child can easily divide 12 by 3.
All that remains is to add up the results, i.e. 72:3=10 (obtained when 30 was divided by 3) + 10 (30 divided by 3) + 4 (12 divided by 3).
72:3=24
We did not use long division, but the child understood the reasoning and completed the calculations without difficulty.
After simple examples, you can move on to studying long division and teach your child to correctly write examples in a “corner”. To begin with, use only examples of division without a remainder.
How to explain long division to a child: solution algorithm
Large numbers are difficult to divide in your head; it is easier to use column division notation. To teach your child to perform calculations correctly, follow the algorithm:
- Determine where the dividend and divisor are in the example. Ask your child to name the numbers (what we will divide by what).
213:3
213 - dividend
3 - divider
- Write down the dividend - "corner" - divisor.
- Determine which part of the dividend we can use to divide by a given number.
We reason like this: 2 is not divisible by 3, which means we take 21.
- Determine how many times the divisor “fits” in the selected part.
21 divided by 3 - take 7.
- Multiply the divisor by the selected number, write the result under the “corner”.
7 multiplied by 3 - we get 21. Write it down.
- Find the difference (remainder).
At this stage of reasoning, teach your child to check himself. It is important that he understands that the result of a subtraction must ALWAYS be less than the divisor. If it doesn't work out, you need to increase the selected number and perform the action again.
- Repeat the steps until the remainder is 0.
How to reason correctly to teach a 2-3 grade child to divide by column
How to explain division to a child 204:12=?
1.
Write it down in a column.
204 is the dividend, 12 is the divisor.
2.
2 is not divisible by 12, so we take 20.
3.
To divide 20 by 12, take 1. Write 1 under the “corner”.
4.
1 multiplied by 12 gets 12. We write it under 20.
5.
20 minus 12 gets 8.
Let's check ourselves. Is 8 less than 12 (divisor)? Ok, that's right, let's move on.
6.
Next to 8 we write 4. 84 divided by 12. How much should we multiply 12 to get 84?
It’s hard to say right away, we’ll try to use the selection method.
Let's take 8, for example, but don't write them down yet. We count verbally: 8 multiplied by 12 equals 96. And we have 84! Doesn't fit.
Let's try smaller ones... For example, let's take 6. We check ourselves verbally: 6 multiplied by 12 equals 72. 84-72=12. We got the same number as our divisor, but it should be either zero or less than 12. So the optimal number is 7!
7.
We write 7 under the “corner” and perform the calculations. 7 multiplied by 12 gives 84.
8.
We write the result in a column: 84 minus 84 equals zero. Hooray! We decided correctly!
So, you have taught your child to divide by column, now all that remains is to practice this skill and bring it to automatism.
Why is it difficult for children to learn long division?
Remember that problems with mathematics arise from the inability to quickly do simple arithmetic operations. IN primary school you need to practice and make addition and subtraction automatic, and learn the multiplication table from cover to cover. All! The rest is a matter of technique, and it is developed with practice.
Be patient, do not be lazy, once again explain to the child what he did not learn in the lesson, tediously but meticulously understand the reasoning algorithm and talk through each intermediate operation before voicing a ready answer. Give additional examples to practice skills, play math games- this will bear fruit and you will see the results and rejoice at your child’s success very soon. Be sure to show where and how you can apply the acquired knowledge in everyday life.
Dear readers! Tell us how you teach your children to do long division, what difficulties you have encountered and how you have overcome them.
At school these actions are studied from simple to complex. Therefore, it is imperative to thoroughly understand the algorithm for performing these operations on simple examples. So that later there will be no difficulties with dividing decimal fractions into a column. After all, this is the most difficult version of such tasks.
This subject requires consistent study. Gaps in knowledge are unacceptable here. Every student should learn this principle already in the first grade. Therefore, if you miss several lessons in a row, you will have to master the material on your own. Otherwise, later problems will arise not only with mathematics, but also with other subjects related to it.
Second required condition successful study mathematics - move on to examples of long division only after addition, subtraction and multiplication have been mastered.
It will be difficult for a child to divide if he has not learned the multiplication table. By the way, it is better to teach it using the Pythagorean table. There is nothing superfluous, and multiplication is easier to learn in this case.
How are natural numbers multiplied in a column?
If difficulty arises in solving examples in a column for division and multiplication, then you should begin to solve the problem with multiplication. Since division is the inverse operation of multiplication:
- Before multiplying two numbers, you need to look at them carefully. Choose the one with more digits (longer) and write it down first. Place the second one under it. Moreover, the numbers of the corresponding category must be under the same category. That is, the rightmost digit of the first number should be above the rightmost digit of the second.
- Multiply the rightmost digit of the bottom number by each digit of the top number, starting from the right. Write the answer below the line so that its last digit is under the one you multiplied by.
- Repeat the same with another digit of the lower number. But the result of multiplication must be shifted one digit to the left. In this case, its last digit will be under the one by which it was multiplied.
Continue this multiplication in a column until the numbers in the second factor run out. Now they need to be folded. This will be the answer you are looking for.
Algorithm for multiplying decimals
First you have to imagine that the data are not decimals, but natural. That is, remove the commas from them and then proceed as described in the previous case.
The difference begins when the answer is written down. At this moment, it is necessary to count all the numbers that appear after the decimal points in both fractions. This is exactly how many of them need to be counted from the end of the answer and put a comma there.
It is convenient to illustrate this algorithm using an example: 0.25 x 0.33:
Where to start learning division?
Before solving long division examples, you need to remember the names of the numbers that appear in the long division example. The first of them (the one that is divided) is divisible. The second (divided by) is the divisor. The answer is private.
After this, using a simple everyday example, we will explain the essence of this mathematical operation. For example, if you take 10 sweets, then it’s easy to divide them equally between mom and dad. But what if you need to give them to your parents and brother?
After this, you can get acquainted with the rules of division and master them in specific examples. First simple ones, and then move on to more and more complex ones.
Algorithm for dividing numbers into a column
First, let us present the procedure for natural numbers divisible by a single-digit number. They will also be the basis for multi-digit divisors or decimal fractions. Only then should you make small changes, but more on that later:
- Before doing long division, you need to figure out where the dividend and divisor are.
- Write down the dividend. To the right of it is the divider.
- Draw a corner on the left and bottom near the last corner.
- Determine the incomplete dividend, that is, the number that will be minimal for division. Usually it consists of one digit, maximum of two.
- Choose the number that will be written first in the answer. It should be the number of times the divisor fits into the dividend.
- Write down the result of multiplying this number by the divisor.
- Write it under the incomplete dividend. Perform subtraction.
- Add to the remainder the first digit after the part that has already been divided.
- Choose the number for the answer again.
- Repeat multiplication and subtraction. If the remainder is zero and the dividend is over, then the example is done. Otherwise, repeat the steps: remove the number, pick up the number, multiply, subtract.
How to solve long division if the divisor has more than one digit?
The algorithm itself completely coincides with what was described above. The difference will be the number of digits in the incomplete dividend. Now there should be at least two of them, but if they turn out to be less than the divisor, then you have to work with the first three digits.
There is one more nuance in this division. The fact is that the remainder and the number added to it are sometimes not divisible by the divisor. Then you have to add another number in order. But the answer must be zero. If you are dividing three-digit numbers into a column, you may need to remove more than two digits. Then a rule is introduced: there should be one less zero in the answer than the number of digits removed.
You can consider this division using the example - 12082: 863.
- The incomplete dividend in it turns out to be the number 1208. The number 863 is placed in it only once. Therefore, the answer is supposed to be 1, and under 1208 write 863.
- After subtraction, the remainder is 345.
- You need to add the number 2 to it.
- The number 3452 contains 863 four times.
- Four must be written down as an answer. Moreover, when multiplied by 4, this is exactly the number obtained.
- The remainder after subtraction is zero. That is, the division is completed.
The answer in the example would be the number 14.
What if the dividend ends in zero?
Or a few zeros? In this case, the remainder is zero, but the dividend still contains zeros. There is no need to despair, everything is simpler than it might seem. It is enough to simply add to the answer all the zeros that remain undivided.
For example, you need to divide 400 by 5. The incomplete dividend is 40. Five fits into it 8 times. This means that the answer should be written as 8. When subtracting, there is no remainder left. That is, the division is completed, but a zero remains in the dividend. It will have to be added to the answer. Thus, dividing 400 by 5 equals 80.
What to do if you need to divide a decimal fraction?
Again, this number looks like a natural number, if not for the comma separating the whole part from the fractional part. This suggests that the division of decimal fractions into a column is similar to that described above.
The only difference will be the semicolon. It is supposed to be put in the answer as soon as the first digit from the fractional part is removed. Another way to say this is this: if you have finished dividing the whole part, put a comma and continue the solution further.
When solving examples of long division with decimal fractions, you need to remember that any number of zeros can be added to the part after the decimal point. Sometimes this is necessary in order to complete the numbers.
Dividing two decimals
It may seem complicated. But only at the beginning. After all, how to divide a column of fractions by a natural number is already clear. This means that we need to reduce this example to an already familiar form.
It's easy to do. You need to multiply both fractions by 10, 100, 1,000 or 10,000, and maybe by a million if the problem requires it. The multiplier is supposed to be chosen based on how many zeros are in the decimal part of the divisor. That is, the result will be that you will have to divide the fraction by a natural number.
And this will be the worst case scenario. After all, it may happen that the dividend from this operation becomes an integer. Then the solution to the example with division into a column of fractions will be reduced to the very simple option: operations with natural numbers.
As an example: divide 28.4 by 3.2:
- They must first be multiplied by 10, since the second number has only one digit after the decimal point. Multiplying will give 284 and 32.
- They are supposed to be separated. Moreover, the whole number is 284 by 32.
- The first number chosen for the answer is 8. Multiplying it gives 256. The remainder is 28.
- The division of the whole part has ended, and a comma is required in the answer.
- Remove to remainder 0.
- Take 8 again.
- Remainder: 24. Add another 0 to it.
- Now you need to take 7.
- The result of multiplication is 224, the remainder is 16.
- Take down another 0. Take 5 each and you get exactly 160. The remainder is 0.
The division is complete. The result of example 28.4:3.2 is 8.875.
What if the divisor is 10, 100, 0.1, or 0.01?
Just like with multiplication, long division is not needed here. It is enough to simply move the comma in the desired direction for a certain number of digits. Moreover, using this principle, you can solve examples with both integers and decimal fractions.
So, if you need to divide by 10, 100 or 1,000, then the decimal point is moved to the left by the same number of digits as there are zeros in the divisor. That is, when a number is divisible by 100, the decimal point must move to the left by two digits. If the dividend is a natural number, then it is assumed that the comma is at the end.
This action gives the same result as if the number were to be multiplied by 0.1, 0.01 or 0.001. In these examples, the comma is also moved to the left by a number of digits equal to the length of the fractional part.
When dividing by 0.1 (etc.) or multiplying by 10 (etc.), the decimal point should move to the right by one digit (or two, three, depending on the number of zeros or the length of the fractional part).
It is worth noting that the number of digits given in the dividend may not be sufficient. Then the missing zeros can be added to the left (in the whole part) or to the right (after the decimal point).
Division of periodic fractions
In this case, it will not be possible to obtain an accurate answer when dividing into a column. How to solve an example if you encounter a fraction with a period? Here we need to move on to ordinary fractions. And then divide them according to the previously learned rules.
For example, you need to divide 0.(3) by 0.6. The first fraction is periodic. It converts to the fraction 3/9, which when reduced gives 1/3. The second fraction is the final decimal. It’s even easier to write it down as usual: 6/10, which is equal to 3/5. The rule for dividing ordinary fractions requires replacing division with multiplication and divisor with the reciprocal. That is, the example comes down to multiplying 1/3 by 5/3. The answer will be 5/9.
If the example contains different fractions...
Then several solutions are possible. Firstly, common fraction You can try to convert it to decimal. Then divide two decimals using the above algorithm.
Secondly, every final decimal fraction can be written as a common fraction. But this is not always convenient. Most often, such fractions turn out to be huge. And the answers are cumbersome. Therefore, the first approach is considered more preferable.