Dividing natural numbers: rules, examples, solutions. Properties of division of natural numbers
Division is an arithmetic operation inverse to multiplication, through which one finds out how many times one number is contained in another.
The number being divided is called divisible, the number being divided by is called divider, the result of division is called private.
Just as multiplication replaces repeated addition, division replaces repeated subtraction. For example, dividing the number 10 by 2 means finding out how many times the number 2 is contained in 10:
10 - 2 - 2 - 2 - 2 - 2 = 0
By repeating the operation of subtracting 2 from 10, we find that 2 is contained in 10 five times. This can be easily checked by adding 2 times five or multiplying 2 by 5:
10 = 2 + 2 + 2 + 2 + 2 = 2 5
To record division, use the sign: (colon), ÷ (obelus) or / (slash). It is placed between the dividend and the divisor, with the dividend written to the left of the division sign and the divisor to the right. For example, writing 10: 5 means that the number 10 is divisible by the number 5. To the right of the division record, put an = (equal) sign, after which the result of the division is written. Thus, the complete division notation looks like this:
This entry reads like this: the quotient of ten and five equals two, or ten divided by five equals two.
Division can also be considered as the action by which one number is divided by as many equal parts, how many units are contained in another number (by which it is divided). This determines how many units are contained in each individual part.
For example, we have 10 apples, dividing 10 by 2 we get two equal parts, each containing 5 apples:
Checking division
To check division, you can multiply the quotient by the divisor (or vice versa). If the result of multiplication is a number equal to the dividend, then the division is correct.
Consider the expression:
where 12 is the dividend, 4 is the divisor, and 3 is the quotient. Now let's check the division by multiplying the quotient by the divisor:
or divisor by quotient:
Division can also be checked by division; to do this, you need to divide the dividend by the quotient. If the result of division is a number equal to the divisor, then the division is performed correctly:
The main property of the private
The quotient has one important property:
The quotient will not change if the dividend and the divisor are multiplied or divided by the same natural number.
For example,
32: 4 = 8, (32 3) : (4 3) = 96: 12 = 8 32: 4 = 8, (32: 2) : (4: 2) = 16: 2 = 8
Dividing a number by itself and one
For any natural number a the following equalities are true:
a : 1 = a
a : a = 1
Number 0 in division
When zero is divided by any natural number, the result is zero:
0: a = 0
You cannot divide by zero.
Let's look at why you can't divide by zero. If the dividend is not zero, but any other number, for example 4, then dividing it by zero would mean finding a number that, when multiplied by zero, results in the number 4. But there is no such number, because any number, when multiplied by zero, gives again zero.
If the dividend is also equal to zero, then division is possible, but any number can serve as a quotient, because in this case any number after multiplication by the divisor (0) gives us the dividend (i.e., 0 again). Thus, division, although possible, does not lead to a single definite result.
Division natural numbers
A lesson in the integrated application of knowledge and methods of action
based on the system-activity teaching method
5th grade
Full name Zhukova Nadezhda Nikolaevna
Place of work : MAOU secondary school No. 6 Pestovo
Job title : math teacher
Topic Division of natural numbers
(training session on the integrated application of knowledge and methods of action)
Target: creating conditions for improving knowledge and skillsand skills in dividing natural numbers and methods of action in modified conditionsand non-standard situations
UDD:
Subject
They simulate a situation, illustrating the arithmetic operation and the progress of its execution, select an algorithm for solving a non-standard problem, and solve equations based on the relationship between the components and the result of the arithmetic operation.
Metasubject
Regulatory : define the goal educational activities, implement the means to achieve it.
Cognitive : Convey content in compressed or expanded form.
Communication: they know how to express their point of view, trying to substantiate it, giving arguments.
Personal:
They explain to themselves their individual immediate goals of self-development, give a positive self-assessment of the result of educational activities, understand the reasons for the success of educational activities, demonstrate cognitive interest to study the subject.
Lesson progress
1. Organizational moment.
In work we use addition,
Honor and honor to the addition!
Let's add patience to skills,
And the amount will bring success.
Don't forget subtraction.
So that the day is not wasted,
From the sum of efforts and knowledge
We will subtract idleness and laziness!
Multiplication will help in work,
To useful work was,
Let's multiply hard work a hundredfold
Our deeds will increase.
Division serves in practice,
It will always help us.
Who shares the difficulties equally?
Share the successes of labor!
Any of the following will help:
They bring us good luck.
And that’s why we’re together in life
Science and labor are advancing.
II. Formulating the topic and objectives of the lesson
Did you like the poem? What did you like about it?
(students' answers)
You said it very well. The lines we read fit very well with our lesson today. Remember a poem you heard and try to determine topic of the lesson.
(Division of natural numbers) (slide 1) . Write down the date and topic of the lesson in your notebook.
Today is the first lesson on the topic “Dividing numbers”? What else are you not good at and what would you like to learn? (students' answers)
So, today we will improve our division skills, learn to justify our decisions, find errors and correct them, evaluate our work and the work of our classmates.
III. Preparation for active educational and cognitive activities
- Motivation for schoolchildren's learning
Humanity has been learning division for the longest time. To this day, the saying “Division is a difficult thing” has been preserved in Italy. This is difficult both from the point of view of mathematics, and technically, and morally. Not every person is given the ability to divide and share.
In the Middle Ages, a person who mastered division received the title “doctor of the abacus”
Abacus is an abacus.
At first there was no sign for the division action. This action was written in words.
And Indian mathematicians wrote division with the first letter of the name of the action.
The colon sign for division came into use in 1684 thanks to the German mathematician Gottfried Wilhelm Leibniz.
Division is also indicated by an oblique or horizontal line. This sign was first used by the Italian scientist Fibonacci.
- How do we divide multi-digit numbers? (Corner)
Do you remember what components are called when divided?(slide 2)
- Do you know that the components of division: dividend, divisor, quotient were first introduced in Russia by Magnitsky. Who is this scientist and what was this scientist’s real name? Prepare answers to these questions for the next lesson.
2) Update background knowledge students
- Graphic dictation
1. Division is an action by which the product and one of the factors are used to find another factor.
2. Division has a commutative property.
3.To find the dividend, you need to multiply the quotient by the divisor.
4. You can divide by any number.
5.To find the divisor, you need to divide the dividend by the quotient.
6. An equality with a letter whose value must be found is called an equation
(Designation: yes; - no) (slide 3)
KEY: (slide 4)
B) Individual work of students using cards.
(simultaneously with dictation)
- Prove that the number 4 is the root of the equation 44: x + 9 = 20.
- Solution . If x=4 then 44:4+9=20
11+9=20
20=20, that's right.
2. Calculate: a) 16224: 52 = (312) d) 13725: 45 = (305)
B) 4230:18 = (235) d) 54756: 39 = (1404)
c) 9800: 28= (350)
3. Solve the equation: 124: (y – 5) = 31
Answer: y=9
4. Two students work using cards: solve 3 tasks each and ask each other theory questions
c) Collective verification individual work(slide 5)
(Students ask the answering questions about theory)
- Application of knowledge and methods of action
A) Independent work with self-test(Slides 6 -7)
Select and solve only those examples in which the quotient has three digits:
Option 1 Option 2
A)2888: 76 = (38) a)2491:93= (47)
B)6539:13 = (503) b)5698: 14= (407)
B) 5712: 28 = (204) c) 9792: 32 = (306)
B) Physical education minute.
They stood up together and stretched.
Hands on the belt, turned around.
Right, left, once, twice,
They turned their heads.
We stood on our toes,
The back was held with a string
Now, sit down quietly,
We haven't done everything yet.
B) Work in pairs (slide 8)
(during work in pairs, if necessary, the teacher gives consultations)
No. 484 (textbook, page 76)
X cm is the length of one of the sides of the octagon
4x+4 4 =24
4x+16=24
4x=24-16
4x=8
X=2
2 cm is the length of one of the sides of the octagon
Solve equations:
a) 96: x = 8 b) x: 60 = 14 c) 19 * x = 76
D) Work in groups
Before you start completing tasks, read the rules for working in groups
Group I (1st row)
Rules for working in groups
Correct errors:
A)9100:10=91; a) 9100:10 = 910
B)5427: 27=21; b) 5427: 27 = 201
B)474747: 47=101; c) 474 747: 47 = 10101
D)42·11=442. d) 42 11 = 462
Group II (2nd row)
Rules for working in groups
- Actively participate in collaboration.
- Listen carefully to your interlocutor.
- Do not interrupt your friend until he finishes his story.
- Express your point of view on this issue, while being polite.
- Don't laugh at other people's shortcomings and mistakes, but tactfully point them out.
Check if the task was completed correctly. Offer your solution
Find the value of the expression x:19 +95 if x =1995.
Solution.
If x=1995, then x:19 +95 = 1995:19 +95=15+95=110
(1995: 19 + 95 = 200)
Group III (3rd row)
Rules for working in groups
- Actively participate in collaboration.
- Listen carefully to your interlocutor.
- Do not interrupt your friend until he finishes his story.
- Express your point of view on this issue, while being polite.
- Don't laugh at other people's shortcomings and mistakes, but tactfully point them out.
Prove that an error was made in solving the equation.
Solve the equation.
124: (y-5) =31
U-5 = 124·31 y – 5 =124: 31
U-5 = 3844 y – 5 = 4
Y = 3844+ 5 y = 4+ 5
Y = 3849 y = 9
Answer: 3849 Answer: 9
D) Mutual check of work in pairs
Students exchange notebooks and check each other's work, highlight errors with a simple pencil and mark them
E) Group report on the work done
(Slides 5-7)
The slide shows the task for each group. The group leader explains the mistake made and writes the group's proposed solution on the board.
V. Monitoring student knowledge
Individual testing “Moment of Truth”
Test on the topic “Division”
Option1
1.Find the quotient of 2876 and 1.
a) 1; b) 2876; c) 2875; d) your answer_______________
2.Find the root of equation 96: x =8
a) 88; b) 12; c) 768; d) your answer ________________
3 .Find the quotient of 3900 and 13.
a) 300; b) 3913; c) 30; d) your answer_______________
4 .One box contains 48 pencils, and the other contains 4 times less. How many pencils are there in two boxes?
a) 192; b) 60; c) 240; d) your answer________________
5. Find two numbers if one of them is 3 times larger than the other, and their
Their sum is 32.
a) 20 and 12; b) 18 and 14; c)26 and 6; d) your answer_________
Test on the topic “Division”
Last name, first name___________________________________________
Option 2
Underline the correct answer or write down your answer.
1 .Find the quotient of 2563 and 1.
a) 1; b) 2563; c) 2564; d) your answer_______________
2. Find the root of Equation 105: x = 3
a) 104; b) 35; c) 315; d) your answer ________________
3 .Find the quotient of 7800 and 13.
a)600; b) 7813; c) 60; d) your answer_______________
4 . In one tub the beekeeper had 24 kg. honey, and in the other 2 times more. How many kilograms of honey did the beekeeper have in two tubs?
a) 12; b) 72; c) 48; d) your answer_______________
5. Find two numbers if one of them is 4 times less than the other, and
Their difference is 27
A) 39 and 12; b) 32 and 8; c) 2 and 29; d) your answer_____________
Test verification key
Option 1
Job number | |||||
9; 36 |
VI. Lesson summary. Homework.
House. Exercise. P.12, No. 520,523,528 (essay).
So, our lesson has come to an end. I would like to interview you about the results of your work.
Continue the sentences:
I am... satisfied/not satisfied with my work in class
I did it...
It was difficult...
The lesson material was... useful/useless for me
What does mathematics teach?
Subject: Division of natural numbers (grade 5) teacher Tatyana Golikova
Georgievna
Target: repeat the method of solving examples by division, table
multiplication, properties of division, rules of division by digit unit,
types of angles, “what does it mean to solve an equation,” finding unknowns
elements of the equation;
develop mathematical speech, attentiveness, outlook,
cognitive activity, ability to analyze, do
assumptions, justify them, classify them;
instilling skills and abilities practical application mathematics,
drawing skills;
development logical thinking, ability to analyze dependence
between values, positive perception of Ukrainian
maintaining health, the ability to evaluate one’s knowledge, creating a situation
success, the feeling of “I CAN”, “I CAN DO EVERYTHING”,
increasing self-esteem, developing internal activity through
emotions and comprehension of the material, awareness of the importance of knowledge in life
person.
Lesson type: practicing skills and abilities
Methods: explanatory - illustrative, gaming, interactive
Forms: heuristic conversation, pair work, mutual control, work in small groups, “I myself - all together”, role play
Equipment: interactive whiteboard, cards different types, marker,
7 sheets of A4, color-coded, tape.
Lesson Plan
1. Spiritual - aesthetic 2 min
2. Motivational 3min
3. Checking homework 5 min
5. Physical education minute 3 min
7. Homework2min
8. Reflection 4min
9.Evaluative 4min
1 Spiritual - aesthetic
All the children stood up quickly.
Good afternoon, please sit down
In order to get ready for work, I suggest repeating the multiplication table
Pick up a pencil, a card and solve the proposed examples in 1.5 minutes, and then read the words in ascending order of numbers.
Find which number “escaped” from the series of natural numbers?
Let's check in unison. The teacher calls the number, and the students call the word.
6:3=2 27:9=3 16:4=4
To drive ships
30:6=5 42:6=7 72:9=8 36:4=9
To fly into the sky
30:3=10 44:4=11 36:3=12
You need to know a lot
26:2=13 42:3=14 150:10=15
There's a lot to know.
Let this quatrain be the motto of today's lesson
2. Motivational
I propose to solve the puzzle in Ukrainian
LEDINE, NILDIK, KASCHAT, TOKBUDO
How many semantic groups can these concepts be divided into?
(You must receive two answer options and justify them)
Topic of today's lesson DIVISION
We opened our notebooks and wrote down the number, great job
3. Checking homework. Updating knowledge
We swapped notebooks and checked “dear colleagues”
Are there any who have not completed the work?
Who found more than two errors?
Thanks to the inspectors, return the notebooks to your neighbors.
What rule did you encounter when performing d/z?
What other properties can you name?
4.1 task 1
I suggest you go on a trip "In the Animal World"
Take the example cards and solve them in your notebooks. Please note that not all examples are solved in writing; division by digit unit is encountered.
The work is given 4-5 minutes. After completion, the teacher accepts the answers, checking them with the corresponding group and writes with a marker on the sheets. Groups answer in any order. The teacher suggests arranging the sheets in the right order to get a story (The sheets are ordered like a RAINBOW)
Red Orange Yellow Green
1) 13000:1000; 1)120000:1000; 1) 300000:10000; 1) 35000:100;
2) 432:24; 2) 476:28; 2) 960:64; 2) 4485:23;
3) 11092:47 3) 6765:123. 3) 7956:234 3) 2790:62.
Light Blue Blue Purple
1) 43000:1000; 1) 11000:100; 1) 1400000:100000;
2) 1856:64 ; 2) 1734:34; 2) 5166:63;
3) 9126:234. 3) 3608:164. 3) 3210:214.
Gorilla sleeps 13000:1000= 13 hours a day, every day 432:24=18 hours a day, and in a state of hibernation, a hedgehog can survive without food 11092:47=236 days
Orange
The speed of the fish is the sword 120000:1000
120km/h, and the speed of the perch
476:28=17 km/h, and the speed of a shark 6765: 12355 km/h
Horses live up to 300000:10000=30 years, and dogs up to 960:64=15 years old, and the dog's life record is 7956:234=34 years old
Weight polar bear reaches 35000:100=350kg, blue whale up to 4485:23=195 tons, and the weight of the East European Shepherd 2790:62=45kg
In humans normal temperature body 36.6 0 , the highest of all warm-blooded pigeons and ducks, up to 43000:1000=43 0 , and the lowest is in the anteater 1856:64=29 0 , dog body temperature 9126:234= 39 0 .
Grape snail survives 11000:100=110 0 frost, but dies when 1734:34= 51 0 heat. Comfortable air temperature for humans 3608:164=22 0
Violet
Length of a large anaconda found in South America, can reach 1400000:100000=14m and in diameter 5166:63= 82cm. And the buildings of African termite warriors reach a height 3210:214=15m
4.2 task 2.
It's okay if we don't know the answer to a question. The main thing is to want to find the answer. We have already told you that if you are sick or miss a lesson for any reason, or something doesn’t work out for you, we have a wonderful TEXTBOOK assistant! We will now solve equations; if someone has forgotten how to find an unknown element of an equation, then do not be lazy to read page 124 of the textbook
Solve equations No. 470(3,4,6)
At the window No. 470(3)
Medium №470(4)
At the door No. 470(6)
Using the representative from the series, equations are solved. Additional task, for those who quickly mastered the equation “I AM WELL DONE! »
“I’M GREAT! » (10x-4x)∙21=2268.
№470(3) №470(4) №470(6)
I'm great!
11x+6x=408; 33m- m=1024 ; 476:x=14 (10x-4x)∙21=2268.
x=24m=32 x=34 x=18
Keys to equations
X=204, P=32, M=304, !=18; Yu=302, A=34, U=24, K=3.
The correct answers are “HURRAY!”
5. Physical education minute
We're tired of sitting,
You just need a little bit of reading.
Hands up, hands down,
Marvel at the susida!
Hands up, hands on hips,
І earn some skoki.
Shvidko sat down and sat down.
The legs became dull.
Splash at the valley once.
For work. Everything is great!
They straightened their backs and put their hands on the desk.
To organize attention, the game “CORNERS”
Show an acute angle, right angle, obtuse angle, developed angle, 30 0, 70 0, 97 0, 150 0, etc., rhumb?
Problem No. 487
We read, draw up a diagram, analyze, find a solution, write down.
Let's see what's happening on the slide
Let's stage it with the students.
Making a table
| 24 km less |
|||
1) 58∙4=232(km) the first train traveled
2) 232+24=256(km) the second train traveled
3) 256:4=64(km/h)
Answer: the second train was traveling at a speed of 64 km/h
7. Homework
Can you handle this task at home? Let's write down the d/z.
No. 488, No. 471 (II column), repeat the rules for solving equations, creative task(rhumb)
8. Reflection
Game of Know and Dunno
Znayka asks Dunno about the properties of division, the rules for finding the elements of an equation, how the quotient will change if...
And Dunno answers!
We had some unused leaves on the table. They show dots. What type of work is this like? (graphic dictation)
How many dots are there on the piece of paper? How many questions will there be? I remind you of the answers
"yes"; "No" ; not sure
· · · · · · · ·
1. Numbers when divided are called dividend, divisor, quotient
2. I realized that division is not at all difficult
3. To find an unknown divisor, you need to divide the dividend by the quotient
4. To find an unknown factor, you need to divide the product by the known factor
5. Today in class I was interested.
6. I worked conscientiously in class.
7. I'm proud of myself.
The assistants collect cards in a row, and the teacher announces the marks.
| 1) 13000:1000; 2) 432:24; 3) 11092:47. | 1) 13000:1000; 2) 432:24; 3) 11092:47. | 1) 13000:1000; 2) 432:24; 3) 11092:47. | 1) 13000:1000; 2) 432:24; 3) 11092:47. |
| 1)120000:1000; 2) 476:28; 3) 6765:123. | 1)120000:1000; 2) 476:28; 3) 6765:123. | 1)120000:1000; 2) 476:28; 3) 6765:123. | 1)120000:1000; 2) 476:28; 3) 6765:123. |
| 1) 300000:10000; 2) 960:64; 3) 7956:234. | 1) 300000:10000; 2) 960:64; 3) 7956:234. | 1) 300000:10000; 2) 960:64; 3) 7956:234. | 1) 300000:10000; 2) 960:64; 3) 7956:234. |
| 1) 35000:100; 2) 4485:23; 3) 2790:62. | 1) 35000:100; 2) 4485:23; 3) 2790:62. | 1) 35000:100; 2) 4485:23; 3) 2790:62. | 1) 35000:100; 2) 4485:23; 3) 2790:62. |
| 1) 43000:1000; 2) 1856:64; 3) 9126:234. | 1) 43000:1000; 2) 1856:64; 3) 9126:234. | 1) 43000:1000; 2) 1856:64; 3) 9126:234. | 1) 43000:1000; 2) 1856:64; 3) 9126:234. |
| 1) 11000:100; 2) 1734:34; .3) 3608:164. | 1) 11000:100; 2) 1734:34; .3) 3608:164. | 1) 11000:100; 2) 1734:34; .3) 3608:164. | 1) 11000:100; 2) 1734:34; .3) 3608:164. |
| 1) 1400000:100000; 2) 5166:63; 3) 3210:214. | 1) 1400000:100000; 2) 5166:63; 3) 3210:214. | 1) 1400000:100000; 2) 5166:63; 3) 3210:214. | 1) 1400000:100000; 2) 5166:63; 3) 3210:214. |
| 1) 13000:1000; 2) 432:24; 3) 11092:47. | 1)120000:1000; 2) 476:28; 3) 6765:123. | 1) 300000:10000; 2) 960:64; 3) 7956:234. | 1) 35000:100; 2) 4485:23; 3) 2790:62. |
| 1) 1400000:100000; 2) 5166:63; 3) 3210:214. | 1) 11000:100; 2) 1734:34; .3) 3608:164. | 1) 43000:1000; 2) 1856:64; 3) 9126:234. |
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Single-digit natural numbers are easy to divide in your head. But how to divide multi-digit numbers? If a number already has more than two digits, mental counting can take a lot of time, and the likelihood of errors when operating with multi-digit numbers increases.
Column division is a convenient method often used for dividing multi-digit natural numbers. It is this method that this article is devoted to. Below we will look at how to perform long division. First, let's look at the algorithm for dividing a multi-digit number by a single-digit number into a column, and then - multi-digit by multi-digit number. In addition to theory, the article provides practical examples of long division.
Yandex.RTB R-A-339285-1
It is most convenient to keep notes on squared paper, since when making calculations, the lines will prevent you from getting confused in the digits. First, the dividend and divisor are written from left to right in one line, and then separated by a special division sign in a column, which looks like:
Let's say we need to divide 6105 by 55, let's write:
We will write intermediate calculations under the dividend, and the result will be written under the divisor. In general, the column division scheme looks like this:
Please remember that calculations will require free space on the page. Moreover, than more difference in the dividend and divisor digits, the more calculations there will be.
For example, to divide the numbers 614,808 and 51,234 you will need less space, than for dividing the number 8058 by 4. Despite the fact that in the second case the numbers are smaller, the difference in the number of their digits is greater, and the calculations will be more cumbersome. Let's illustrate this:
It is most convenient to practice practical skills on simple examples. Therefore, let's divide the numbers 8 and 2 into a column. Of course, this operation is easy to perform in your head or using the multiplication table, but a detailed analysis will be useful for clarity, even though we already know that 8 ÷ 2 = 4.
So, first we write down the dividend and divisor according to the column division method.
The next step is to find out how many divisors the dividend contains. How to do this? We successively multiply the divisor by 0, 1, 2, 3. . We do this until the result is a number equal to or greater than the dividend. If the result immediately results in a number equal to the dividend, then under the divisor we write the number by which the divisor was multiplied.
Otherwise, when we get a number greater than the dividend, under the divisor we write the number calculated at the penultimate step. In place of the incomplete quotient we write the number by which the divisor was multiplied at the penultimate step.
Let's go back to the example.
2 · 0 = 0 ; 2 · 1 = 2 ; 2 · 2 = 4 ; 2 · 3 = 6 ; 2 4 = 8
So, we immediately got a number equal to the dividend. We write it under the dividend, and write the number 4, by which we multiplied the divisor, in the place of the quotient.
Now all that remains is to subtract the numbers under the divisor (also using the column method). In our case, 8 - 8 = 0.
This example- division of numbers without a remainder. The number obtained after subtraction is the remainder of the division. If it is equal to zero, then the numbers are divided without a remainder.
Now let's look at an example where numbers are divided with a remainder. Divide the natural number 7 by the natural number 3.
IN in this case, sequentially multiplying three by 0, 1, 2, 3. . we get as a result:
3 0 = 0< 7 ; 3 · 1 = 3 < 7 ; 3 · 2 = 6 < 7 ; 3 · 3 = 9 > 7
Under the dividend we write the number obtained in the penultimate step. Using the divisor we write down the number 2 - the incomplete quotient obtained in the penultimate step. It was by two that we multiplied the divisor when we got 6.
To complete the operation, subtract 6 from 7 and get:
This example is dividing numbers with a remainder. The partial quotient is 2 and the remainder is 1.
Now, after considering elementary examples, let's move on to dividing multi-digit natural numbers into single-digit ones.
We will consider the column division algorithm using the example of dividing the multi-digit number 140288 by the number 4. Let us say right away that it is much easier to understand the essence of the method using practical examples, and this example was not chosen by chance, as it illustrates all the possible nuances of dividing natural numbers in a column.
1. Write the numbers together with the division symbol in a column. Now look at the first digit on the left in the dividend notation. Two cases are possible: the number defined by this digit is greater than the divisor, and vice versa. In the first case, we work with this number, in the second, we additionally take the next digit in the dividend notation and work with the corresponding two-digit number. In accordance with this point, let’s highlight in the example record the number with which we will work initially. This number is 14 because the first digit of the dividend 1 is less than the divisor 4.
2. Determine how many times the numerator is contained in the resulting number. Let's denote this number as x = 14. We successively multiply the divisor 4 by each member of the series of natural numbers ℕ, including zero: 0, 1, 2, 3 and so on. We do this until we get x or a number greater than x as a result. When the result of multiplication is the number 14, we write it under the highlighted number according to the rules for writing subtraction in a column. The factor by which the divisor was multiplied is written under the divisor. If the result of multiplication is a number greater than x, then under the highlighted number we write the number obtained at the penultimate step, and in place of the incomplete quotient (under the divisor) we write the factor by which the multiplication was carried out at the penultimate step.
In accordance with the algorithm we have:
4 0 = 0< 14 ; 4 · 1 = 4 < 14 ; 4 · 2 = 8 < 14 ; 4 · 3 = 12 < 14 ; 4 · 4 = 16 > 14 .
Under the highlighted number we write the number 12 obtained in the penultimate step. In place of the quotient we write the factor 3.
3. Subtract 12 from 14 using a column, write the result under the horizontal line. By analogy with the first point, we compare the resulting number with the divisor.
4. Number 2 less number 4, therefore we write down under the horizontal line after the two the number located in the next digit of the dividend. If there are no more digits in the dividend, then the division operation ends. In our example, after the number 2 obtained in the previous paragraph, we write down the next digit of the dividend - 0. As a result, we note a new working number - 20.
Important!
Points 2 - 4 are repeated cyclically until the end of the operation of dividing natural numbers by a column.
2. Let's count again how many divisors are contained in the number 20. Multiplying 4 by 0, 1, 2, 3. . we get:
Since we received a number equal to 20 as a result, we write it under the marked number, and in place of the quotient, in the next digit, we write 5 - the multiplier by which the multiplication was carried out.
3. We carry out the subtraction in a column. Since the numbers are equal, the result is the number zero: 20 - 20 = 0.
4. We will not write down the number zero, since this stage is not the end of division. Let’s just remember the place where we could write it down and write next to it the number from the next digit of the dividend. In our case, the number is 2.
We take this number as a working number and again carry out the steps of the algorithm.
2. Multiply the divisor by 0, 1, 2, 3. . and compare the result with the marked number.
4 0 = 0< 2 ; 4 · 1 = 4 > 2
Accordingly, under the marked number we write the number 0, and under the divisor in the next digit of the quotient we also write 0.
3. Perform the subtraction operation and write the result under the line.
4. To the right under the line add the number 8, since this is the next digit of the number being divided.
Thus, we get a new working number - 28. We repeat the points of the algorithm again.
Having done everything according to the rules, we get the result:
Move it below the line last digit dividend - 8. We repeat algorithm points 2 - 4 for the last time and get:
In the very bottom line we write the number 0. This number is written only at the last stage of division, when the operation is completed.
Thus, the result of dividing the number 140228 by 4 is the number 35072. This example has been analyzed in great detail, and when solving practical tasks there is no need to describe all the actions so thoroughly.
We will give other examples of dividing numbers into a column and examples of writing solutions.
Example 1. Column division of natural numbers
Divide the natural number 7136 by the natural number 9.
After the second, third and fourth steps of the algorithm, the record will take the form:
Let's repeat the cycle:
The last pass, and we read the result:
Answer: The partial quotient of 7136 and 9 is 792 and the remainder is 8.
When deciding practical examples Ideally, do not use explanations in the form of verbal comments at all.
Example 2. Dividing natural numbers into a column
Divide the number 7042035 by 7.
Answer: 1006005
The algorithm for dividing multi-digit numbers into a column is very similar to the previously discussed algorithm for dividing a multi-digit number by a single-digit number. To be more precise, the changes concern only the first point, while points 2 - 4 remain unchanged.
If, when dividing by a single-digit number, we looked only at the first digit of the dividend, now we will look at as many digits as there are in the divisor. When the number determined by these digits is greater than the divisor, we take it as the working number. Otherwise, we add another digit from the next digit of the dividend. Then we follow the steps of the algorithm described above.
Let's consider the application of the algorithm for dividing multi-digit numbers using an example.
Example 3. Dividing natural numbers into a column
Let's divide 5562 by 206.
The divisor contains three signs, so let’s immediately select the number 556 in the dividend.
556 > 206, so we take this number as a working number and move on to point 2 of the agloritm.
Multiply 206 by 0, 1, 2, 3. . and we get:
206 0 = 0< 556 ; 206 · 1 = 206 < 556 ; 206 · 2 = 412 < 556 ; 206 · 3 = 618 > 556
618 > 556, so under the divisor we write the result of the penultimate action, and under the dividend we write the factor 2
Perform column subtraction
As a result of subtraction we have the number 144. To the right of the result, under the line, we write the number from the corresponding digit of the dividend and get a new working number - 1442.
We repeat points 2 - 4 with him. We get:
206 5 = 1030< 1442 ; 206 · 6 = 1236 < 1442 ; 206 · 7 = 1442
Under the marked working number we write 1442, and in the next digit of the quotient we write the number 7 - the multiplier.
We carry out subtraction into a column, and we understand that this is the end of the division operation: there are no more digits in the divisor to write to the right of the subtraction result.
To conclude this topic, we will give another example of dividing multi-digit numbers into a column, without explanation.
Example 5. Column division of natural numbers
Divide the natural number 238079 by 34.
Answer: 7002
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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 numbers three 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 by 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 division. large numbers. If prime numbers like 16 and 4, can be divided, and the answer is clear - 4. That 512:8 in the mind is not easy for a child. 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. Largest number, divisible by 8 without a remainder to 51 – 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 by 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
Intermediate level
Difficult level
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