How to multiply three-digit numbers by two-digit numbers. Rules for multiplying two-digit numbers in a column. Motivation for learning activities

How to multiply by column

Multiplication of multi-digit numbers is usually performed in a column, writing the numbers under each other so that the digits of the same digits are under each other (units under units, tens under tens, etc.). For convenience, the number that has more digits is usually written on top. An action sign is placed on the left between the numbers. A line is drawn under the multiplier. The numbers of the product are written below the line as they are obtained.

Let's first consider multiplying a multi-digit number by a single-digit number. Let's say you need to multiply 846 by 5:

Multiplying 846 by 5 means adding 5 numbers, each of which is equal to 846. To do this, it is enough to first take 5 times 6 units, then 5 times 4 tens and finally 5 times 8 hundreds.

5 times 6 units = 30 units, i.e. 3 tens. We write 0 under the line in place of units, and remember 3 tens. For convenience, so as not to remember, you can write 3 above the tens of the multiplicand:

5 times 4 tens = 20 tens, add to them 3 more tens = 23 tens, i.e. 2 hundreds and 3 tens. We write 3 tens under the line in the place of tens, and remember 2 hundreds:

5 times 8 hundreds = 40 hundreds, add another 2 hundreds = 42 hundreds. We write 42 hundreds under the line, i.e. 4 thousand and 2 hundreds. Thus, the product of 846 by 5 turns out to be equal to 4230:

Now let's look at the multiplication of multi-digit numbers. Let's say we need to multiply 3826 by 472:

Multiplying 3826 by 472 means adding 472 identical numbers, each of which is equal to 3826. To do this, you need to add 3826 first 2 times, then 70 times, then 400 times, i.e. multiply the multiplicand separately by the digit of each digit of the multiplier and the resulting products add up into one sum.

2 times 3826 = 7652. We write the resulting product under the line:

This is not the final product as long as we have only multiplied by one digit of the multiplier. The resulting number is called partial product. Now our task is to multiply the multiplicand by the tens digit. But before that, you need to remember one important point: each partial product must be written under the number by which the multiplication occurs.

Multiply 3826 by 7. This will be the second partial product (26782):

We multiply the multiplicand by 4. This will be the third partial product (15304):

We draw a line under the last partial product and add all the resulting partial products. We get the full product (1 805 872):

If a zero is found in the multiplier, then usually they do not multiply by it, but immediately move on to the next digit of the multiplier:

When the multiplicand and (or) multiplier end in zeros, the multiplication can be performed without paying attention to them, and at the end, add as many zeros to the product as there are in the multiplicand and the multiplier together.

For example, you need to calculate 23,000 · 4500. First, multiply 23 by 45, ignoring the zeros:

And now, on the right, we will add as many zeros to the resulting product as there are in the multiplicand and in the multiplier together. The result is 103,500,000.

Column multiplication calculator

This calculator will help you perform multiplication by column. Simply enter the multiplicand and multiplier and click the Calculate button.

>> Lesson 13. Multiplying by a three-digit number

Let's write the numbers in a column (one below the other). The top line is the larger number, the bottom line is the smaller number.

The rightmost digit (sign) of the top number must be above the rightmost digit of the bottom number. On the left side between the numbers we put an action sign. For us it is “×” (multiplication sign).
First, multiply the entire top number by the last digit of the bottom number. The result is written below the line below the rightmost number.

Multiply the number from above by digit (sign) from right to left.

We got a number greater than or equal to “10”.

Therefore, only the last digit of the result goes under the line. This is "2". The number of tens of the work (we have “4 tens”) is placed above the neighbor to the left of “7”.
Multiply "2" by "6".

The result of multiplication by the second digit must be written under the second digit of the result of the first multiplication operation.

Now having mastered multiplication by column, you can multiply arbitrarily large numbers.

COLUMN MULTIPLICATION OF TWO-DIGIT NUMBERS

Math trainer

The program is a math simulator for consolidating skills multiplying two-digit numbers with a column.

There are 20 examples to solve. Two random two-digit numbers need to be multiplied by a column.

To go to the beginning of solving examples, press the “START” button

In the upper left part of the math simulator page, the number of examples that remain to be solved is indicated.

On the right side of the page is an example that needs to be solved. On the left side the same example is written in a column.

Use the cursor keys to move up/down/right/left across the cells. Press buttons 0-9 on the keyboard and enter intermediate answers and the final answer.

If the example is solved correctly, 5 points are awarded. If you give the correct answer three times in a row, a bonus is awarded.

For an incorrect answer, 3 points are deducted.

Errors made during the calculation are corrected in red. It will be immediately clear at what stage of the calculations the error was made.

The final page of the math simulator presents the results: the number of points, errors, bonuses.

If at multiplication by column mistakes were made; examples in which they occurred will be listed below.

Rules for multiplying two-digit numbers in a column

Method multiplication by column, allows you to simplify multiplication of numbers. Column multiplication involves sequential multiplication first number, to all digits of the second number, subsequent addition of the resulting products, taking into account indentation, depending on the position of the digit of the second number.

Let's look at how to multiply by column using the example of finding the product of two numbers 625 × 25 .

With a larger number of digits in the second number, we get that our products are lined up on the right in the form of a “ladder”.

4 As a result of multiplication we get 2 works, 3125 And 1250 , we will sequentially add their numbers together from right to left, in the order in which they appear, and write the result of their addition below. If the sum of the digits during addition exceeds 9 , then divide the amount by 10 , we write the remainder of the division under the current numbers, and move the whole part of the division to the left.

As a result we get .

The most important rule with which we begin to study multiplication by column:

Column multiplication by a two-digit number

Example: 46 times 73

This example can be written in a column.

Under the number 46 we write the number 73 according to the rule:

Units are written under units, and tens are written under tens.

1 We start multiplying with units.

Multiply 3 by 6. You get 18.

• 18 units is 1 ten and 8 ones.
• We write 8 ones under the units, and remember 1 ten and add them to the tens.

Now let's multiply 3 by 4 tens. It turns out 12.

12 tens, and 1 more, for a total of 13 tens.

There are no hundreds in this example, so we immediately write 1 in place of hundreds.

138 is first incomplete work.

2 Multiplying tens.

7 tens times 6 ones equals 42 tens.

7 tens multiplied by 4 tens equals 28 hundreds. 28 hundreds, and 4 more makes 32 hundreds.

There are no thousands in this example, so I immediately write 3 in place of thousands.

3220 is second incomplete work.

3 We add the first and second incomplete products according to the rule of addition in a column.

How to quickly multiply two-digit numbers in your head?

How to quickly multiply large numbers, how to master such useful skills? Most people find it difficult to verbally multiply two-digit numbers by single-digit numbers. And there is nothing to say about complex arithmetic calculations. But if desired, the abilities inherent in every person can be developed. Regular training, a little effort and the use of effective techniques developed by scientists will allow you to achieve amazing results.

Choosing traditional methods

Methods of multiplying two-digit numbers that have been proven for decades do not lose their relevance. The simplest techniques help millions of ordinary schoolchildren, students of specialized universities and lyceums, as well as people engaged in self-development, improve their computing skills.

Multiplication using number expansion

The easiest way to quickly learn to multiply large numbers in your head is to multiply tens and units. First, the tens of two numbers are multiplied, then the ones and tens alternately. The four numbers received are summed up. To use this method, it is important to be able to remember the results of multiplication and add them in your head.

For example, to multiply 38 by 57 you need:

• factor the number into (30+8)*(50+7) ;
• 30*50 = 1500 – remember the result;
• 30*7 + 50*8 = 210 + 400 = 610 – remember;
• (1500 + 610) + 8*7 = 2110 + 56 = 2166

Naturally, it is necessary to have excellent knowledge of the multiplication table, since it will not be possible to quickly multiply in your head in this way without the appropriate skills.

Multiplication by column in the mind

Many people use a visual representation of the usual columnar multiplication in calculations. This method is suitable for those who can memorize auxiliary numbers for a long time and perform arithmetic operations with them. But the process becomes much easier if you learn how to quickly multiply two-digit numbers by single-digit numbers. To multiply, for example, 47*81 you need:

• 47*1 = 47 – remember;
• 47*8 = 376 – remember;
• 376*10 + 47 = 3807.

Speaking them out loud while simultaneously summing them up in your head will help you remember intermediate results. Despite the difficulty of mental calculations, after a short practice this method will become your favorite.

The above multiplication methods are universal. But knowing more efficient algorithms for some numbers will greatly reduce the number of calculations.

Multiply by 11

This is perhaps the simplest method that is used to multiply any two-digit numbers by 11.

It is enough to insert their sum between the digits of the multiplier:
13*11 = 1(1+3)3 = 143

If the number in brackets is greater than 10, then one is added to the first digit, and 10 is subtracted from the amount in brackets.
28*11 = 2 (2+8) 8 = 308

Multiplying large numbers

It is very convenient to multiply numbers close to 100 by decomposing them into their components. For example, you need to multiply 87 by 91.

• Each number must be represented as the difference between 100 and one more number:
  (100 - 13)*(100 - 9)
  The answer will consist of four digits, the first two of which are the difference between the first factor and the subtracted from the second bracket, or vice versa - the difference between the second factor and the subtracted from the first bracket.
  87 – 9 = 78
  91 – 13 = 78
• The second two digits of the answer are the result of multiplying those subtracted from two parentheses. 13*9 = 144
• The result is the numbers 78 and 144. If, when writing down the final result, a number of 5 digits is obtained, the second and third digits are summed. Result: 87*91 = 7944 .

These are the simplest methods of multiplication. After using them multiple times, bringing the calculations to automation, you can master more complex techniques. And after a while, the problem of how to quickly multiply two-digit numbers will no longer worry you, and your memory and logic will improve significantly.

Mathematics lesson on the topic “Multiplying three-digit numbers in a column.” 3rd grade

A bad teacher presents the truth, a good teacher teaches you to find it.

The goal of modern Russian education has become the full formation and development of the student’s abilities to independently outline an educational problem, formulate an algorithm for solving it, control the process and evaluate the result.
The new standard is distinguished by the implementation of a system-activity approach to teaching, where the student’s position is active, where he acts as an initiator and creator, and not a passive performer.

UUD formed in the lesson:

Personal:

• understanding the student’s internal position at the level of a positive attitude towards the lesson
• moral and ethical assessment of acquired content
• adherence to moral standards and ethical requirements in behavior
• self-assessment based on success criteria

Communication:

• planning educational cooperation with the teacher and peers
• expressing your thoughts with sufficient completeness and accuracy, using criteria to justify your judgment

Cognitive:

• extracting necessary information from tasks
• setting and formulating the problem
• identification of primary and secondary information
• putting forward hypotheses and their substantiation

Regulatory:

• self-organization and organization of your workplace
• exercising self-control
• recording individual difficulties in a trial educational action, ability to predict

I. Organizational moment ( Presentation– slide 1)

Checking readiness for the lesson (slide 2)

– Check how your “workplace”, textbook, pencil case is organized.
- Let's do some finger exercises. (children touch their finger to their neighbor on the desk and say):

I wish (thumb)
Large (medium)
Success (index)
In everything (nameless)
And everywhere (little finger)
Good luck! (whole palm)

Motivation for learning activities.

– I also want to wish you good luck.
-Where do we start our work?

1. Encrypted word

– I offer you a very interesting task!
- What should be done?

Annex 1 (work in pairs)

- What word did you get? (Success)
– Good luck and success await each of you in class today!
– Name the largest three-digit number. (124 ) (slide 3)
- Tell me everything you know about this number. (It is natural, not round, it is in 124th place in the series of natural numbers, it is preceded by the number 123, followed by the number 125. The sum of the digits of this number is 7. It is three-digit. It contains 1 hundred, 2 tens, 4 units)

2. Writing a number as a sum of digit terms

– Write it down as a sum of digit terms: 124 = 100 + 20 + 4 (slide 4)
– Swap notebooks with your deskmate and check each other’s work.
– Now tell me, what do we know (can) about three-digit numbers?

II. Motivation

I know (I can) (slide 4)

• read
• write down
• compare
• represented as a sum of bit terms
• perform oral addition and subtraction techniques
• perform oral multiplication and division techniques

– What skills did we use when completing this task with the number 124? (Expand three-digit numbers into the sum of their digit terms)
– Where can we use these skills? (When solving examples, for ease of calculation)
- Look at the blackboard.

800*3 200*4
412*2 123*3
112*4 300*3

– What two groups can these expressions be divided into? (Expressions for multiplying round and non-round three-digit numbers)
– Which column example can we solve easily and quickly? Why? (First, we know how to multiply round numbers)
– Write down the answers to the examples in the first column in your notebook.
– Whoever wrote it down, sit up straight. Check the sample. (Slide 5)
– Look at the examples in the second column. Can we solve these examples right away? Why? (No, we can't)

I want to know (slide 6)

– Would you like to know how to solve such examples? (How to multiply three-digit numbers in a column)
– Formulate the topic of today’s lesson.

“Multiplying three-digit numbers in a column” (slide 7)

– What goals can we set? (Learn to multiply three-digit numbers in a column)
- Yes, that's right. You are not yet familiar with multiplying three-digit numbers in a column!
– This is our main goal in the lesson!
– Make your guesses, how will we multiply a three-digit number by a one-digit number?

III. Finding a solution

– What can help us so as not to make mistakes in solving examples? (NEED ALGORITHM!)
– Now you need to work and correctly arrange the order of actions in the algorithm.
– You and I will divide into two groups.
– The first group must restore the sequence of the algorithm, as you would act when multiplying.
– With the second group we will verbally analyze the algorithm of actions.
– The guys from the second group will evaluate the correctness of your algorithm. (Children line up in the right order)
– Read your algorithms, and now compare them with the one on my slide. (slide 8)

ALGORITHM

1. I’M WRITING.
2. MULTIPLY THE UNITS.
3. WE WRITE UNITS UNDER UNITS.
4. MULTIPLYING TENS.
5. WE WRITE TENS UNDER THE TENS.
6. MULTIPLY HUNDREDS.
7. WE WRITE HUNDREDS UNDER HUNDREDS.
8. READING THE ANSWER.

IV. Primary consolidation

– Now let’s use the algorithm and solve the examples of the second column (at the board with an explanation)

412 * 2 = 824
123 * 3 = 369
112 * 4 = 448

– Did you like solving the examples?
– Now let’s rest a little.

IV. Fizminutka (slide 9)

– I will give tasks, and you will give the answer using the number of movements:

SO MANY TIMES STOMP YOUR FOOT - 12: 3
SO MANY TIMES WE CLAP YOUR HANDS - 25: 5
WE WILL COME SO MANY TIMES - 36: 9
WE LEAN NOW - 18: 3
WE WILL JUMP EXACTLY THIS MUCH - 36: 6
- ARE YOU RESTED? ON THE ROAD AGAIN.

V. Solution of the problem

– Can you use the skills acquired in class when solving problems?
- Then we decide!

(slide 10)

“The age of the birch tree under which the travelers built their hut is 121 years, and the age of the oak tree that grows nearby is 3 times older. How old is the oak tree? How many years older is oak than birch?
1) 121 * 3 = 363 (years) – the age of the oak.
2) 363 - 121 = 242 (g.) – the difference.

Answer: The age of the oak is 363 years; the oak is 242 years older than the birch.

V. Independent work (slide 11)

– Can you solve the examples on your own?

223 * 3
212 * 4
241 * 2
313 * 3
413 * 2

– Exchange notebooks and check whether your neighbor solved the examples correctly.

VII. Reflection on learning activities in the lesson and lesson summary

– What was our goal at the beginning of the lesson?
- Did you manage?

Found out (algorithm for multiplying three-digit numbers into a column) (slide 12)

– Where will this knowledge be useful to you? (At home, in a store.)
- Let's see how we worked, how you assessed our work and the work of the class.
– Now onto the “mood ladder” (slide 13) Attach your star to the step that corresponds to your feelings, mood, state of your soul that you had throughout the lesson.

Multiplying natural numbers in a column, examples, solutions.

It is convenient to multiply natural numbers in a special way, which is called “ multiplication by column" or " multiplication by column" The beauty of this method is that the multiplication of multi-digit natural numbers is reduced to the sequential multiplication of two single-digit numbers.

In this article we will analyze in detail the algorithm for multiplying two natural numbers by a column. We will describe the sequence of actions step by step, while simultaneously showing the solutions to the examples.

Page navigation.

What do you need to know to multiply natural numbers by column?

Intermediate calculations when multiplying by column are carried out using the multiplication table, so it is advisable to know it by heart so as not to waste time searching for the desired result.

Sooner or later, when multiplying with a column, we will be faced with multiplying a single-digit natural number by zero. In this case, we will use the property of multiplying a natural number by zero: a·0=0, Where a– an arbitrary natural number..

We recommend that you understand the material in the article column addition. This is due to the fact that at one of the stages of columnar multiplication it is necessary to add intermediate results (which are called incomplete products) using the principle of columnar addition.

Writing factors when multiplying in a column.

Let's start with the rules for writing factors when multiplying by a column.

The second multiplier is written below the first multiplier so that the first digits on the right other than the digit 0 , are located one below the other. A horizontal line is drawn below the written factors, and a multiplication sign of the form “×” is placed on the left. Here are examples of how to correctly write factors when multiplying in columns. The entries in the column of products of numbers are shown below 352 And 71 , 550 And 45 002 , and 534 000 And 4 300 .



We've sorted out the recording.

Now you can proceed directly to the process of multiplying two natural numbers in a column. First, let's look at multiplying a multi-digit number by a single-digit number. After this, we will analyze the multiplication by a column of two multi-digit natural numbers.

Column multiplication of a multi-digit natural number by a single-digit number.

Now we will give column multiplication algorithm multi-digit natural number to a single-digit natural number. We will do this while simultaneously describing the solution to the example.

Suppose we need to multiply a given multi-digit natural number 45 027 for a given single digit number 3 .

We write the factors in the same way as multiplication by a column (in this case, the single-digit number appears under the rightmost sign of the multi-digit number).

For our example, the entry will look like this:


Now we multiply the units digit of a given multi-digit number by a given single-digit number. If we get a number less than 10 , then we write it under the horizontal line in the same column in which the given single-digit number to be multiplied is located. If we get the number 10 or a number greater than 10 , then under the horizontal line we write down the value of the units digit of the resulting number, and remember the value of the tens digit (we will add the remembered number to the result of the multiplication in the next step, after which we will delete the remembered number from memory).

That is, we multiply 7 (this is the value of the units digit of the first multiplier 45 027 ) on 3 . We get 21 . Because 21 more 10 , then write the number under the line 1 (this is the value of the units digit of the resulting number 21 ) and remember the number 2 (this is the value of the tens place of the number 21 ). At this step, the entry will look like this:


We move on to the next stage of the column multiplication algorithm. We multiply the value of the tens place of a given multi-digit number by a given single-digit number and add to the product the number memorized at the previous stage (if we memorized it). If the result is a number less than ten, then we write it under the horizontal line to the left of the number already written there. If the result is the number ten or a number greater than ten, then under the horizontal line we write down the value of the units digit of the resulting number, and remember the value of the tens digit (we also use it in the next step).

So let's multiply 2 (this is the value of the tens place of the first multiplier 45 027 ) on 3 , we have 6 . To this number we add the number remembered in the previous step 2 , we get 6+2=8 . Because 8 less than 10 , then write the number under the horizontal line 8 to the desired position (in this case, we do not need to remember any number, that is, now we have no numbers in memory). We have:


At the next step, we proceed in a similar way, but we already multiply the value of the hundreds place of a given multi-digit number by a given single-digit natural number. We add the remembered number to this product (if it was remembered); compare the result with the number 10 ; if necessary, remember the new number and write the required number under the horizontal line to the left of the numbers already there.

Multiply 0 on 3 , we get 0 . Since we do not have any number in memory, then to the resulting number 0 no need to add anything. Number 0 less 10 , so we write 0 under the horizontal line at the desired position:


After this, we proceed to multiplying the value of the next digit of a given multi-digit natural number and a given single-digit natural number. We proceed in a similar way until we multiply the values ​​of all digits of a given multi-digit number by a given single-digit natural number.

So let's multiply 5 on 3 , we get 15 . Because 15>10 , then we write below the line 5 and remember the number 1 :


Finally, we multiply 4 on 3 , we get 12 . TO 12 add the number remembered at the previous stage 1 , we have 12+1=13 . Because 13 more than 10 , then write down the number 3 to the right place and remember the number 1 :


Note that if at the last stage we had to remember a number, then it needs to be written under the horizontal line to the left of the numbers already there.

We have a number in our memory 1 , so it needs to be written in the right place under the line:


This completes the process of multiplying a multi-digit natural number by a single-digit natural number with a column, and the result of the multiplication is the number written under the horizontal line.

Thus, multiplication by a column of natural numbers 45 027 And 3 led us to the result 135 081 .

For clarity, let us schematically depict the algorithm for multiplying a multi-digit natural number by a single-digit natural number with a column (this figure reflects only the general picture, but does not show all the nuances).



It remains to deal with multiplication by a column of a multi-digit natural number, in the notation of which there is a digit on the right 0 or several numbers 0 in a row, by a single digit number. We will also consider all the steps of column multiplication in such cases using an example. Moreover, let’s take the numbers from the previous example, but add several digits to the notation for a multi-digit number 0 on right.

So, let's multiply the natural numbers 4 502 700 (we added two numbers 0 ) per number 3 .

In this case, we first write down the numbers to be multiplied in the same way as multiplication by a column would suggest:


After this, we carry out multiplication in a column as if numbers 0 on the right there is no.

Let's use the result from the example already solved above:


At the final stage of multiplication, in a column under the horizontal line, to the right of the digits already there, we write down as many digits 0 , how many of them are on the right in the original number being multiplied.

In our example, you need to add two numbers 0 . The entry will look like:


This completes the multiplication by column.

The result of multiplying a multi-digit natural number 4 502 700 , the entry of which ends in zeros, to a single-digit natural number 3 is 13 508 100 .

Column multiplication of two multi-digit natural numbers.

Let us describe all the stages of the algorithm for multiplying two multivalued natural numbers in a column.

We will carry out the description together with the solution of the example. Now we will assume that in the records of multiplied natural numbers there are no digits on the right 0 . We will consider the multiplication of multi-valued natural numbers whose records end in zeros at the end of this paragraph.

Multiply numbers by column 207 on 8 063 .

We start by writing the factors one below the other. Note that it is more convenient to place a multiplier on top, the entry of which consists of a larger number of characters (in our example, we will write the number on top 8 603 , since in his entry 4 sign, and the number 207 three-digit). If the records of the factors contain the same number of characters, then it does not matter which of the factors is written on top. So, we place the factors one below the other so that the numbers of the first factor are under the numbers of the second factor from right to left:


Now at each next step we will receive the so-called incomplete works.

The first stage of the algorithm is to multiply the first factor by a column (in our example this is the number 8 063 ) to the value of the units digit of the second factor (in our example, the value of the units digit of the number 207 is the number 7 ). All actions are similar to multiplying a multi-digit number by a single-digit number with a column (if necessary, return to the previous paragraph of this article), as a result, under the horizontal line we have the first incomplete product. At this stage, the record will take the following form:


Let's move on to the second stage. At this stage, we multiply the first factor with a column (in our example this is the number 8 063 ) by the value of the tens place of the second multiplier, if it is not equal to zero. If the value of the tens place of the second multiplier is zero, then we proceed to the next stage (in our example, the value of the tens place of the number 207 equals zero, so we move on to the third stage). We write the results below the line below the number already written there, starting from the position that corresponds to the tens place.

At the third, fourth and so on stages, we act in a similar way, multiplying the first factor (the number 8 063 ) to the value of the hundreds place of the second multiplier (if it is not equal to zero), then to the value of the thousands place (if it is not equal to zero) and so on. We write the results below the line below the numbers already written there, starting from the position corresponding to the digit of the single-digit number by which the multiplication is carried out at this stage.

So let's multiply the number 8 063 to the value of the hundreds place of a number 207 , that is, by number 2 . We obtain the second incomplete product, and the solution to the example will take the following form:


So, all incomplete products have been calculated. The last stage of the algorithm remains, at which all incomplete products are added up, and this is done in the same way as when adding in a column. Addition is performed using an existing record (incomplete products remain in the places where they are written, that is, they do not move anywhere), another horizontal line is drawn below, a “+” sign is placed on the left, and the addition results are written under the bottom line . If there is only one number in the column, and there is no number stored in the memory at the previous stage, then it is written under the horizontal line.

In our example we get:


The number formed below is the result of multiplying the original multi-digit natural numbers. So, the product of numbers 8 063 And 207 equals 1 669 041 .

For clarity, let us schematically depict the process of multiplying two natural numbers with a column.



Let us show the solution to another example for securing the material.

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• Law of St. Petersburg dated May 31, 2010 N 273-70 “On administrative offenses in St. Petersburg” (Adopted by the Legislative Assembly of St. Petersburg on May 12, 2010) (with amendments and additions) Law of St. Petersburg dated May 31, 2010 N 273-70 "On administrative [...]
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At school these actions are studied from simple to complex. Therefore, it is imperative to thoroughly understand the algorithm for performing these operations using 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.

The second prerequisite for successfully studying mathematics is to 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:

1. 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.
2. 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.
3. 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 need to imagine that the given fractions are not decimals, but natural ones. 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. That's exactly how many of them you need to count 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 become familiar with the division rules and master them using 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 you do 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 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 column division of fractions will be reduced to the simplest 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.
• Carry 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, you can try to convert a common fraction to a 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.