Column division rules for multi-digit numbers. Teaching schoolchildren long division

Amazing discovery made by our reader. Her son didn’t understand how to do long division in class. Wanting to help her son, she opened the textbook and saw that... she saw nothing. For some reason, there were no explanations for the topic in the book. How to teach a child long division if there is a similar methodological incident in your child’s book?

What you need to know to learn to divide

Mathematics doesn't like gaps. All knowledge must be as strong as bricks. If a child doesn't know the basics, division will be incredibly difficult. What should you pay attention to?

Does the student know the names of the elements when dividing?
Make sure your child doesn't forget the multiplication table.
Repeat the digits of the number.

Let's start dividing

We will look at how to teach a child to divide using concrete examples. Follow the reasoning and be attentive to the numbers.

Separate the dividend from the divisor with a corner bracket.

Let's think about it this way: can 4 be divided by 5? No you can not. Therefore, we take not 4, but 46. Let’s remember the multiplication table (you can take a printout), what number in the multiplication table by 5 is closest to 46? – 45. How many times does 5 fit into 45? – 9 times. We sign 45 to 46, units under units, so as not to get confused. We write nine “on the shelf” - in the corner.

If you subtract 45 from 46, how much do you get? -1. One less than five? - less. So we divided correctly.

One is not divisible by 5, we take away the remaining number - 5, we get 15. Is fifteen divisible by five? - shares. How much is it? – 3. We write three in the corner. We check the solution: three times 5 equals 15. Sign it under the previous number. Subtract fifteen from fifteen and it becomes zero. We used all the numbers from the dividend, which means we solved the example correctly.

In the corner we wrote down two numbers - 9 and 3, we got the number 93. Ninety-three is the quotient, which is the solution to our example.

When explaining to a schoolchild how to learn to divide by column, check reverse action: 93*5. Also, solve more difficult options.

There are other, special cases - you will learn about them from the program. If there really is “nothing” in the textbook, make it a rule to check the solution with your class work. From the class notebook it is easy to understand what method the teacher uses and repeat it when explaining homework.

>> Lesson 13. Division by two digits and three digit number

Divide 876 by 24. Calculating 800: 20 = 40 shows that the answer should be a number close to 40.

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

The number of hundreds 8 is single-digit, so we divide 87 tens by 24. You get 3 tens and another 15 tens remain (87 - 3 24 = 15). 15 tens and 6 units is 156. And if 156 is divided by 24, you get 6 and 12 as a remainder (156 - 24 6 = 12). In total you get 3 tens and 6 units, that is, 36, and the remainder is 12. This is written like this:

10*. Find the sum of all possible two-digit numbers all of whose digits are odd.

Peterson Lyudmila Georgievna. Mathematics. 4th grade. Part 1. - M.: Yuventa Publishing House, 2005, - 64 p.: ill.

Lesson plans for 4th grade mathematics download, textbooks and books for free, development of mathematics lessons online

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Schoolchildren learn column division, or, more correctly, the written technique of corner division, already in the third grade. primary school, but often so little attention is paid to this topic that by grades 9-11 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. Let's check that he less than 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

Let's first consider simple cases division when the quotient produces a single-digit number.

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

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

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

The quotient of 265 and 53 is 5.

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

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

The quotient will be a single digit number.

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

The quotient of 184 and 23 is 8.

Let's consider more complex cases of division.

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

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

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

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

The quotient of 768 and 24 is 32.

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

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

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

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

We write 4 units in the line with the new remainder. This means the next incomplete dividend is 224 units. Let's continue the division. 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

One of the most important parts of teaching your child math operations is learning division operations. prime numbers. To teach division to a child, it is necessary that by the time of learning he has already mastered and well understood such mathematical operations as subtraction and addition.

In addition, it is important to have a clear understanding of the very essence of operations such as division and multiplication. Thus, he must understand that the operation of division involves a method of dividing something into equal parts. Finally, you must also learn multiplication operations and have a good knowledge of the multiplication table.

Learning the operation of dividing into parts

On at this stage It is better to form an understanding that the main thing in the process of division is dividing something into equal parts. The most in a simple way learning this for your child would involve asking him to share a few items between him and family members or friends.

For example, take 6 identical objects and ask your child to divide them into two equal parts. You can complicate the task a little by proposing to divide it not into two, but into three equal parts.

An important point here is to carry out operations to divide even numbers of objects. This action will be useful for further stage when the child needs to understand that division is the inverse of multiplication.

Divide and multiply using the multiplication table

Here it is worth explaining to the child about the inverse action of multiplication, called “division.” Based on the multiplication table, show the learner this relationship between division and multiplication using an example.

For example: 2 times 4 is eight. Here, emphasize that the result of multiplication will be the product of two numbers. It will then be better to illustrate the operation of division by pointing out the action of the inverse operation of multiplication.

Divide the resulting answer “8” by any factor – “4” or “2”; the result will always be the factor that was not used in the operation.

It is also worth teaching to recognize categories that describe division operations, such as “divisor,” “dividend,” and “quotient.” It is important to consolidate this knowledge, they are most necessary for the further learning process!

Separate with a column - quickly and easily

Before you start teaching, you should remember with your child what name each number has during the division operation. The main thing is to learn how to quickly and accurately identify these categories.

An illustrative example:

Let's try to divide 938 by 7. In this example, the number 938 will be the dividend, and the number 7 will be the divisor. As a result of the action, the answer will be called the quotient.

It is necessary to write down the numbers, separating them with a “corner”.
Offer to a student from smallest number of the dividend, choose that which is greater than the divisor. Of the numbers 9, 3, 8, the largest will be number 9. Offer to analyze how many sevens the number 9 can contain. There will be only one correct answer here. The first result is 1.
We draw up the division in a column.

Let's multiply the divisor 7 by 1, the answer will be 7. We enter the resulting result under the first number of our dividend, then subtract it into a column. Thus, from 9 we subtract 7 and the answer is 2. We also write this down.

We see a number that is less than the divisor, so we increase it. To do this, we combine it with the unused number of the dividend, that is, with the number 3. We add 3 to the resulting 2.
Then we analyze how many times the divisor 7 will be contained in the number 23. The answer is 3 times and fix it in the quotient. The result of the product 7 by 3 (21) is entered below in the column under the number 23.
All that remains is to find the last number of the quotient. Using the same algorithm, continues the calculations in the column. Subtracts in column 23-21 and gets the difference, equal to the number 2. Of all the dividends, we only have the unused number 8. We combine it with the result 2, we get 28 as an answer.
In conclusion, we analyze how many times the divisor 7 is contained in the number we received. Correct answer 4 times. We include it in the result. As a result, our answer obtained during the division process is 134.

The most important thing when teaching a child the division method is to master and clearly understand the algorithm of actions, because in fact it is extremely simple.

If your child is excellent at operating the multiplication table, then he should not have any difficulties with “reverse” division. Therefore, it is very important to practice the acquired skills all the time. Don't stop there.

To easily teach a young student the division method, you should:

at the age of three years, correctly grasp the terms “whole” and “part.” An understanding of the concept of the whole, as an inseparable category, must be formed, as well as the perception individual parts the whole in the concept of an independent object.
correctly understand and understand the methods of division and multiplication.

In order for the child to enjoy the lessons, interest in mathematics should be aroused in everyday situations, and not just in the learning process.

Therefore, train your child’s observation skills, come up with analogies for mathematical actions during games, during the construction process, or in simple observations of nature.

Instructions

Before teaching how to divide two-digit numbers, you need to explain to your child that a number is the sum of tens and units. This will save him from future quite a common mistake that many children make. They begin to divide the first and second digits of the dividend and divisor by each other.

First, work from numbers to single digits. This technique is best practiced using knowledge of the multiplication tables. The more such practice there is, the better. The skills of such division should be brought to automatism, then it will be easier for the child to move on to more complex topic a two-digit divisor, which, like the dividend, is the sum of tens and units.

The most common method of dividing two-digit numbers is the brute method, which involves successively dividing numbers from 2 to 9 so that the resulting product equals the dividend. Example: divide 87 by 29. Reason as follows:

29 times 2 equals 54 – not enough;
29 x 3 = 87 – correct.

Draw the student's attention to the second digits (units) of the dividend and divisor, which are convenient to focus on when using the multiplication table. For example, in the above example, the second digit of the divisor is 9. Think about how much you need to multiply the number 9 so that the number of units of the product equals 7? In this case, there is only one answer - 3. This greatly simplifies the task of two-digit division. Test your guess by multiplying the entire number 29.

If the task is completed in writing, then it is advisable to use the column division method. This approach is similar to the previous one except that the student does not need to keep the numbers in his head and do mental calculations. It is better to arm yourself with a pencil or a rough piece of paper for written work.

Sources:

multiplying two-digit numbers by two-digit tables

The topic of dividing numbers is one of the most important in the 5th grade math program. Without mastering this knowledge, further study of mathematics is impossible. Divide numbers happen in life every day. And you shouldn’t always rely on a calculator. To divide two numbers, you need to remember a certain sequence of actions.

You will need

A sheet of paper in a square,
pen or pencil

Instructions

Write down the dividend on one line. Separate them with a vertical line two lines high. Draw a horizontal line under the divisor and dividend perpendicular to the previous line. The quotient will be written to the right under this line. Below and to the left of the dividend, under the horizontal line, write down a zero.

Move the one leftmost, but not yet transferred, digit of the dividend down under the last horizontal line. Mark the transferred digit of the dividend with a dot.

Compare the number under the last horizontal line with the divisor. If the number is less than the divisor then continue from step 4, otherwise go to step 5.