2.4: Some Interesting Facts about Division - Mathematics

• be able to recognize a whole number that is divisible by 2, 3, 4, 5, 6, 8, 9, or 10

Division by 2, 3, 4, and 5

Definition: Division by 2

A whole number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.

The numbers 80, 112, 64, 326, and 1,008 are all divisible by 2 since the last digit of each is 0, 2, 4, 6, or 8, respectively.

The numbers 85 and 731 are not divisible by 2.

Definition: Division by 3

A whole number is divisible by 3 if the sum of its digits is divisible by 3.

The number 432 is divisible by 3 since 4 + 3 + 2 = 9 and 9 is divisible by 3.

(432 div 3 = 144)

The number 25 is not divisible by 3 since (2 + 5 = 7), and 7 is not divisible by 3.

Definition: Division by 4

A whole number is divisible by 4 if its last two digits form a number that is divisible by 4.

The number 31,048 is divisible by 4 since the last two digits, 4 and 8, form a number, 48, that is divisible by 4.

(31048 div 4 = 7262)

The number 137 is not divisible by 4 since 37 is not divisible by 4.

Definition: Division by 5

A whole number is divisible by 5 if its last digit is 0 or 5.

Sample Set A

The numbers 65, 110, 8,030, and 16,955 are each divisible by 5 since the last digit of each is 0 or 5.

Practice Set A

State which of the following whole numbers are divisible by 2, 3, 4, or 5. A number may be divisible by more than one number.

26

2

Practice Set A

81

3

Practice Set A

51

3

Practice Set A

385

5

Practice Set A

6,112

2, 4

Practice Set A

470

2, 5

Practice Set A

113,154

2, 3

Division by 6, 8, 9, 10

Definition: Division by 6

A number is divisible by 6 if it is divisible by both 2 and 3.

The number 234 is divisible by 2 since its last digit is 4. It is also divisible by 3 since 2 + 3 + 4 = 9 and 9 is divisible by 3. Therefore, 234 is divisible by 6.

The number 6,532 is not divisible by 6. Although its last digit is 2, making it divisible by 2, the sum of its digits, 6 + 5 + 3 + 2 = 16, and 16 is not divisible by 3.

Definition: Division by 8

A whole number is divisible by 8 if its last three digits form a number that is divisible by 8.

The number 4,000 is divisible by 8 since 000 is divisible by 8.
The number 13,128 is divisible by 8 since 128 is divisible by 8.
The number 1,170 is not divisible by 8 since 170 is not divisible by 8.

Definition: Division by 9

A whole number is divisible by 9 if the sum of its digits is divisible by 9.

The number 702 is divisible by 9 since 7 + 0 + 2 is divisible by 9.
The number 6588 is divisible by 9 since 6 + 5 + 8 + 8 = 27 is divisible by 9.
The number 14,123 is not divisible by 9 since 1 + 4 + 1 + 2 + 3 = 11 is not divisi­ble by 9.

Definition: Division by 10

A Whole number is divisible by 10 if its last digit is 0.

Sample Set B

The numbers 30, 170, 16,240, and 865,000 are all divisible by 10.

Practice Set B

State which of the following whole numbers are divisible 6, 8, 9, or 10. Some numbers may be divisible by more than one number.

900

6, 9, 10

Practice Set B

6,402

6

Practice Set B

6,660

6, 9, 10

Practice Set B

55,116

6, 9

Exercises

For the following 30 problems, specify if the whole number is divisible by 2, 3, 4, 5, 6, 8, 9, or 10. Write "none" if the number is not divisible by any digit other than 1. Some numbers may be divisible by more than one number.

Exercise (PageIndex{1})

48

2, 3, 4, 6, 8

Exercise (PageIndex{2})

85

Exercise (PageIndex{3})

30

2, 3, 5, 6, 10

Exercise (PageIndex{4})

83

Exercise (PageIndex{5})

98

2

Exercise (PageIndex{6})

972

Exercise (PageIndex{7})

892

2, 4

Exercise (PageIndex{8})

676

Exercise (PageIndex{9})

903

3

Exercise (PageIndex{10})

800

Exercise (PageIndex{11})

223

none

Exercise (PageIndex{12})

836

Exercise (PageIndex{13})

665

5

Exercise (PageIndex{14})

4,381

Exercise (PageIndex{15})

2,195

5

Exercise (PageIndex{16})

2,544

Exercise (PageIndex{17})

5,172

2, 3, 4, 6

Exercise (PageIndex{18})

1,307

Exercise (PageIndex{19})

1,050

2, 3, 5, 6, 10

Exercise (PageIndex{20})

3,898

Exercise (PageIndex{21})

1,621

none

Exercise (PageIndex{22})

27,808

Exercise (PageIndex{23})

45,764

2, 4

Exercise (PageIndex{24})

49,198

Exercise (PageIndex{25})

296,122

2

Exercise (PageIndex{26})

178,656

Exercise (PageIndex{27})

5,102,417

none

Exercise (PageIndex{28})

16,990,792

Exercise (PageIndex{29})

620,157,659

none

Exercise (PageIndex{30})

457,687,705

Exercises for Review

Exercise (PageIndex{31})

In the number 412, how many tens are there?

1

Exercise (PageIndex{32})

Subtract 613 from 810.

Exercise (PageIndex{33})

Add 35, 16, and 7 in two different ways.

(35 + 16) + 7 = 51 + 7 = 58
(35 + (16 + 7) = 35 + 23 = 58

Exercise (PageIndex{34})

Find the quotient (35 div 0), if it exists.

Exercise (PageIndex{35})

Find the quotient. (3654 div 42).

87

OperationRuleExample
Additionn + 0 = n5 + 0 = 5
Subtractionn - 0 = n5 - 0 = 5
Multiplicationn × 0 = 05 × 0 = 0
Division0 ÷ n = 0, when n ≠ 00 ÷ 5 = 0
x ÷ 0 is undefined5 ÷ 0 is undefined
Exponentiation0 n = 00 5 = 0
n 0 = 15 0 =1
Root𕔄 =0𕔄 =0
Logarithmlogb(0) is undefinedundefined
Factorial0! = 10! = 1
Sinesin 0º = 0sin 0º = 0
Cosinecos 0º = 1cos 0º = 1
Tangenttan 0º = 0tan 0º = 0

Today, its difficult to imagine how you could have mathematics without zero.

Four is the only number that has the same number of letters as its meaning
F,O,U,R.

A number is divisible by 9 if the sum of the digits is divisible by 9.
9,18,27,36,45.

The number zero does not have its own Roman numeral
IX,XV,CM,VII.

Think to yourself 'How I wish I could calculate pi' and then count the letters in each of the words of that sentence. You now have a way of remembering the first seven digits of pi: 3.141592.
&pi
There are lots of pi facts on the Pi Day page.

The first number to contain the letter a is 'one thousand'. See A NUMBER for the discussion behind this fact.
1,000

The only number to have its letters in alphabetical order is forty. See Alphanumbetical and try to think of other alphanumbetical mathematical words!
Forty

The numbers on opposite sides of a dice add up to seven.

There are more than 64 squares on a chess board. If you count the squares made up of multiple squares there are 204 altogether. There is one 8x8 square, four 7x7 squares, nine 6x6 squares, 16 5x5 squares, 25 4x4 squares, 36 3x3 squares, 49 2x2 squares and 64 1x1 squares.

A prime number has exactly two factors. Two is the only even prime number and it is also the only prime number not to contain the letter 'e'.
2

It should take no more than 20 moves to solve a Rubiks cube no matter which of the 43 quintillion possible starting positions you begin with.

A dodecahedron has twelve pentagonal sides.
.

A googol is one followed by one hundred zeros. This can be written as:
10 100 .

Not many people appreciate (or understand) the mind-blowing fact that:
e i&pi = -1

Triangles, squares and hexagons are the only regular polygons that tessellate.

Check for yourself at the Tessellations page.

The equals sign was invented in 1557 by Welsh mathematician Robert Recorde. The word 'equal' is from the Latin word aequalis as meaning uniform, identical, or equal.
=

111111111 x 111111111 = 12345678987654321.

These two fractions add up to one and between them they contain all of the digits from nought to nine. It is the only way that this can be done.

[Update: 12th Aug 2019, received a message on Twitter from @ignormatyk saying that this was not the only way and there are others for example: 45/90+138/276 and 38/76+145/290]

Eight comes first if all the numbers are arranged alphabetically. What number would come last?
8

More numbers begin with the digit one than any other digit. This result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers and mathematical constants. This fact was famously attributed to physicist Frank Benford, who stated it in 1938, although it had been previously stated by Simon Newcomb in 1881.

Alice in Wonderland learns that in a class of 23 pupils the probability that two have the same birthday is more than a half. Alice is a fictional character created by author and mathemetician Lewis Carroll (1832-1898).

42 is the answer to the 'Ultimate Question of Life, the Universe, and Everything' according to 'The Hitchhiker's Guide to the Galaxy' created by Douglas Adams.

18 is the only number that is twice the sum of its digits.

Most clocks which have Roman numerals on their face use IIII for four instead of the more familiar IV .

.

There are currently over 7 billion people in the world and the number is growing very quickly. To see just how quickly have a look at our population counter.

Any number to the power zero is 1, and zero to any power is 0. The only unanswered question here is what is zero to the power zero?
0 0

The Fibonacci sequence is a sequence of numbers where each term is the sum of the previous two terms. The first two terms are both one. The ratio of the n th term to the next term gets closer to the golden ratio as n increases.

Air France, Iberia, Ryanair, AirTran, Continental Airlines, and Lufthansa don’t have a row 13 on their airlines because they are aware that many of their passengers consider 13 to be an unlucky number. Many office blocks, office buildings and hotels do not have a 13th floor for the same reason.

The following three consecutive numbers are the lowest that are divisible by cubes other than 1:

1375 1376 1377
(divisible by the cubes of 5, 2 and 3 respectively).

The digits of the number are the same as the digits of the power of ten in these cases:
1.3712885742 = 10 0.13712885742
237.5812087593 = 10 2.375812087593
3550.2601815865 = 10 3.5502601815865
46692.4683 = 10 4.669246833

The polar diameter of the Earth is quite close to (within 0.1%) half a billion inches.

The first time a digit repeats six times in succession in pi is at the 762nd position where you can find six nines in a row. This is known as the Feynman Point.

The term googol (a 1 followed by 100 zeroes) was first used by a 9- year old boy,Milton Sirotta, in 1938.

Trivia submitted by Paul Christian Sarmiento, Philippines

As weird as it may seem at first glance, f(x) = -1/(x+1) and g(x) = x/(x+1) have the same derivative.

Trivia submitted by Shiraz Dagia, Malawi

A number if multiply by 11 is just you bring down the last digits add the digit to the number on it's left and bring down the first number.
Example:
(15)(11)=---
Bring down 5
(15)(11)=--5
Then, 5+1=6
(15)(11)=-65
Last bring down the first digit which is 1
So,(15)(11)=165.

Trivia submitted by Jericho Fernandez, Laguna, Philippines

There is a number smaller than 'a thousand' or 'a hundred and one' with the letter a - it is any number like 0.001 or 'one thousandth' which is smaller than 101 or 1000. These decimals can get infinitely smaller so the proper term for the trivia should be that the smallest whole number not just the smallest number.

Trivia submitted by QA, Makaben

You can remember the value of Pi (3.1415926) by counting each word's letters in
'May I have a large container of coffee?'.

Trivia submitted by Ms. Prescott,

Trivia improves critical thinking.

Trivia submitted by Wisani, Gauteng, South Africa

A googolplex is a googol a googol , or (10 100 ) to the power(10 100 ).

Trivia submitted by Anonymous, Planet Earth

40 when written in words "forty" is the only number with letters in alphabetical order, while one is the only one with letters in reverse order.

Trivia submitted by Dustin Joseph C. Manalo, Bulacan,Philippines

The number 0 is originally called cipher.

Trivia submitted by Paras, Philipines

The billionth digit of Pi (3.1415 . ) is 9.

Trivia submitted by Joy, Manila, Philippines

What is the correct mathematical name of the division bar in a fraction? The answer is VINCULUM.

Trivia submitted by Lovely Tinam-isan, Muntinlupa, Philippines

The term "jiffy" is an actual unit of time which is the 1/100th of a second.

Trivia submitted by Josh Anilov C. Funelas, Manila, Philippines

The value of zero was first used by the ancient Indian mathematician Aryabhata.

Trivia submitted by Wency Orbina, Philippines

2520 is the smallest number that is divisible by 1,2,3,4,5,6,7,8,9 and 10.

Trivia submitted by MathWizard, Phillipines

Moving each letter of the word 'yes' 16 places further up the alphabet produces the word 'oui', the French for 'yes'.

Trivia submitted by Greg Ross, Futility Closet

Forty-two percent of Slovenian two-year-olds know the number two, while only four percent of English two-year-olds do.

Trivia submitted by Francie Diep, Popular Science

The word 'twelve' is worth 12 points in Scrabble. .

Trivia submitted by Greg Ross, Futility Closet

The words 'ace, two, three, four, five, six, seven, eight, nine, ten, jack, queen, king' contain 52 letters. There are 52 cards in a pack (excluding jokers).

Trivia submitted by Greg Ross, Futility Closet

If you square11111111 the answer would be 123456787654321. (count the number of 1s and that's the middle number).

Trivia submitted by Ranz Louie Ricasa, Philippines

The polygon with 1000000 sides is called megagon.

Trivia submitted by Ranz Louie Ricasa, Philippines

Trivia submitted by Ranz Louie Ricasa, Philippines

If you find the difference between the number of edges and the number of faces of the tetrahedron, cube, octahedron and other solid shapes, the results will always be 2.

Trivia submitted by Ranz Louie Ricasa, Olongapo City,Philippines

The second hand on a clock is actually the minute hand.

Trivia submitted by Will, Northampton, England

If you write out pi to two decimal places, backwards it spells “pie”.
Found from buzzfeed.

Trivia submitted by Me, England

The Reuleaux Triangle is a shape of constant width, the simplest and best known such curve other than a circle.

Trivia submitted by Manuel Henryk Fabunan, Philippines

The mathematical name for # (number sign) is octothorpe.

Trivia submitted by Kz Fernandez, Philippines

Can you find two numbers without a final 0 that have a product of 10, 100, 1000, 10000 etc. Here is a method:

5(5)*2(2)=25*4=100
25(5)*4(2)=125*8=1000
125(5)*8(2)=625*16=10000
and so on.

Trivia submitted by Ranz Louie Ricasa, Olongapo City,Philippines

Did you know there are five hundred and twenty-five thousand, six hundred minutes in a year? This special number is the main 'hook' of the song Seasons of Love written for the musical Rent.

In the year 1514 the German artist Albrecht Dürer created an engraving called Melencolia with a magic square in the background. The image below shows an enlargement of the magic square. The date appears in the bottom row of the magic square.

Using consecutive whole numbers and counting rotations and reflections of a given square as being the same there are precisely:

1 magic square of size 3 × 3
880 magic squares of size 4 × 4
275,305,224 magic squares of size 5 × 5.

For the 6×6 case, there are estimated to be approximately 1.77 × 10 19 squares.

This trivia is from the excellent book by Professor Ian Stewart called Cabinet Of Mathematical Curiosities.

Zero is the number with the most names or synonyms. It is also known as nought, naught, ow, nil, zilch, zip, diddly-squat, love and scratch.

Did you know that if you add 429 and 138 the answer is 567? The calculation contains the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9.

[Transum: There are another 335 ways to construct a similar calculation. Have a go at finding them using the Nine Digit Sum drag-and-drop activity.]

Trivia submitted by April Jean Elumba, University Of Southern Mindanao

J and K are the only letters not in any of the numbers when written as words.

Using only addition, you can add 8's to get the number 1,000 by:
888+88+8+8+8=1,000.

Trivia submitted by Nightshade, Iligan City

The word Trivia comes from the Latin meaning three ways (tri is the prefix for three). At a three way junction there would be a signpost giving information about each direction. This information could be called trivia!

There are exactly 8! (eight factorial) minutes in four weeks.

This is calculated as follows: 4x7x24x60

All ten digit pandigital numbers are divisible by 3 (a pandigital number contains all the digits 0 to 9).

[You can find out more about tests for divisibility here.]

King's Cross station in London has a platform zero! It is the longest platform at the station.

The number 2 is the only prime number that doesn't have the letter e in its name.

Trivia submitted by Origaminess, Naperville

Did you know that rather than (in the UK) having 1p, 2p, 5p etc. coins it would be mathematically more efficient to have 1p, 3p, 11p and 37p coins?

Trivia submitted by No Such Thing As A Fish, Podcast

There are 385072 ways of arranging the numbers 1 - 18 in a circle so that the sum of each pair of adjacent numbers is prime.

Try to find just one of them on the Prime Pairs Game page.

153, 370, 371 and 407 are the only three-digit numbers equal to the sum of the cubes of their digits.

Any four digit number typed by using four calculator keys at the corners of a rectangle is a multiple of eleven.

Here's a useful counterintuitive fact: one 18 inch pizza has more 'pizza' than two 12 inch pizzas.
Area of 18" pizza is &pi × 9 2 = 254 square inches.
Area of two 12" pizzas is 2&pi × 6 2 = 226 square inches.

Trivia submitted by Fermat's Library via Twitter,

The volume of a deep-pan pizza with radius Z and depth A is
Pi × Z × Z × A

Older people were born longer ago.

142857 is a circular number
142857 × 2 = 285714
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285.

Trivia submitted by Philip Robinson, New Zealand

The Babylonians were using Pythagoras' Theorem over 1,000 years before Pythagoras was born.

Octothorpe is the another term for the hashtag sign or the no. sign (#).
#octothorpe.

Trivia submitted by Andrew Stevenizer, Philippines

29 is the number of short straight lines needed to make the number 29 when it is written as words:

Here is the formula for solving the equation (ax^2 + bx + c = 0).

Did you know that there is another formula for finding the roots of quadratic equations? It is called the 'citardauq' (the word quadratic backwards) formula and you can read more about it here but you will never need it for school Maths.

206 is the smallest number that when written in words contains all five vowels exactly once:

Did you know that the area of the regular pentagon on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the regular pentagons on the other two sides?

Can you prove that it is true?

A seventh is eleven sevenths of an eleventh.

Trivia submitted by Olberjb, Reddit

You’ve met the first six primes:
2,3,5,7,13,11

They form a nice calculation:

Trivia submitted by Chris Smith @aap03102 Twitter,

12 + 3 - 4 + 5 + 67 + 8 + 9 equals 100.

Trivia submitted by Jimmy Horton, England

The division slash (/) is called the virgule
For example 30/5 = 6.

Trivia submitted by Jimmy Horton, England

All odd numbers have a letter e in them.

Maths teachers are very good correcting their pupils when they use the letter O name when they really mean nought or zero. It is important to distinguish between letters (as used in algebra) and numbers. There are however exceptions to this rule that have come about by common usage:

1. James Bond's number is double 'O' seven
2. The first Scout camp took place in 1907 (nineteen 'O' seven)
3. My telephone number is 'O' two four nine seven one one one
4. A famous TV series was called Hawaii 5 'O'
5. It's my father's birthday. He's celebrating his big five 'O'.

8,549,017,632
contains all of the digits in alphabetical order.

The division symbol is called obelus.

Trivia submitted by Genius Of Sampaguita 2019-2020, TNHS - Main

TWELVE PLUS ONE = ELEVEN PLUS TWO

The left side of this equation is an anagram of the right side!

142,857 × 1 = 142,857
142,857 × 2 = 285,714
142,857 × 3 = 428,571
142,857 × 4 = 571,428
142,857 × 5 = 714,285
142,857 × 6 = 857,142

But how far will this strange sequence of calculations continue?

Here are the only temperatures that are prime integers in both Celsius and Fahrenheit:

-5 o C is equal to 23 o F
5 o C is equal to 41 o F.

Trivia submitted by Fermat's Library On Twitter,

FORTY FIVE is an anagram of OVER FIFTY!

Any six digit number where the first three digits are repeated as the second three digits is always divisible by 7, 11 and 13
eg 136136.

Trivia submitted by Kris Tobin, NSW

41 is prime
41+2 is prime
41+2+4 is prime
41+2+4+6 is prime
41+2+4+6+8 is prime
41+2+4+6+8+10 is prime
41+2+4+6+8+10+12 is prime
41+2+4+6+8+10+12+14 is prime
41+2+4+6+8+10+12+14+16 is prime
41+2+4+6+8+10+12+14+16+18 is prime
41+2+4+6+8+10+12+14+16+18+20 is prime
However, this pattern eventually fails. Do you know when?

Trivia submitted by Math Nerd 1729, Philippines

"A decimal point" is an anagram of "I'm a dot in place".

The ratio of the longer to the shorter side of any A-size paper (A3, A4 etc) is equal to the square root of 2.

The number of milliseconds in a day is:

Only digits 1,2 and 3 are possible in the look and say sequence.
1
11
21
1211
111221
312211
13112221
etc.

Trivia submitted by Mr Phil, TSS

If subtract one from multiplying two successive even numbers, we get a prime number.

Trivia submitted by Vijay Baranwal, Suratgarh, Rajasthan, India

That's 104 interesting and surprising facts but there could be more! Do you know any mathematical trivia that could be added to this page?

Bidmaze

Find your way through the maze encountering mathematical operations in the correct order to achieve the given total. This is an addictive challenge that begins easy but develops into quite a difficult puzzle.

Fraction of .

Practise your ability to find a fraction of a given amount with this self marking exercise. So far this activity has been accessed 41 times and 8 Transum Trophies have been awarded for completing it.

Most Popular Division Worksheets this Week

Division facts worksheets including division tables, division facts and worksheets with individual division facts.

Horizontal division facts worksheets

Manipulatives can help students "get" the concept of division. For example, students could regroup base ten blocks into units, then divide the units into piles. For example, 81 ÷ 9 would end up being 9 piles of 9 units.

Division is essentially asing the question, "How many _____'s are in _____?" For the question, 81 ÷ 9, the prompt would sound like, "How many 9's are in 81?" This prompt will benefit students in later math studies when there are more complex concepts such as dividing decimals or fractions. "How many thirds are in four?" or even better,, How many third cups are in four cups?" If necessary, get out the measuring cups.

Connecting Multiplication and Division Facts

Give students two factors (for example, 3 x 7), ask students to give the product (21) and then ask one student to give a related division fact (for example, 21 ÷ 7 = 3). To keep students attentive, state the factors and have students respond with the product before choosing a student to give the related fact. To keep the pace moving, you can randomly choose students using a deck of name cards.

Part 1 asks students to provide the product of two factors followed by the quotient of its related division fact, using factors less than or equal to 5. Part 2 uses factors up to 10, and the extension offers practice using factors up to 12.

Part 1

Let’s practice our multiplication facts. I’ll give a set of factors, and together, we’ll say the product. Then, a volunteer will give one related division fact. For example, if I say 2 x 4, the product is 8 and one related division fact is 8 ÷ 4 = 2 (or 8 ÷ 2 = 4). Here we go!

• 2 × 5 (10, related division fact is 10 ÷ 5 = 2 or 10 ÷ 2 = 5)
• 3 × 2 (6, related division fact is 6 ÷ 3 = 2 or 6 ÷ 2 =3
• 3 × 3 (9, related division fact is 9 ÷ 3 = 3)
• 2 × 2 (4, related division fact is 4 ÷ 2 = 2)
• 4 × 3 (12, related division fact is 12 ÷ 3 = 4 or 12 ÷ 4 = 3)
• 4 × 4 (16, related division fact is 16 ÷ 4 = 4)
• 5 × 3 (15, related division fact is 15 ÷ 5 = 3 or 15 ÷ 3 = 5)
• 5 × 4 (20, related division fact is 20 ÷ 4 = 5 or 20 ÷ 5 = 4)

While children are enjoying their building of mastery, feel free to repeat. When children are eager for more, try Part 2.

Part 2

Let’s keep working on our related multiplication and division facts, but this time we’ll go even faster. (At some point, you might let the students lead this activity.)

• 5 × 8 (40, related division fact is 40 ÷ 8 = 5 or 40 ÷ 5 = 8)
• 9 × 5 (45, related division fact is 45 ÷ 9 = 5 or 45 ÷ 5 = 9)
• 7 × 6 (42, related division fact is 42 ÷ 6 = 7 or 42 ÷ 7 = 6)
• 6 × 10 (60, related division fact is 60 ÷ 6 = 10 or 60 ÷ 10 = 6)
• 10 × 10 (100, related division fact is 100 ÷ 10 = 10)
• 6 × 8 (48, related division fact is 48 ÷ 8 = 6 or 48 ÷ 6 = 8)
• 9 × 4 (36, related division fact is 36 ÷ 4 = 9 or 36 ÷ 9 = 4)
• 9 × 9 (81, related division fact is 81 ÷ 9 = 9)

As always, when children seem excited for a new challenge, move on.

Extension

Now let’s find some more products and their related division facts.

• 11 × 10 (110, related division fact is 110 ÷ 10 = 11 or 110 ÷ 11 = 10)
• 11 × 8 (88, related division fact is 88 ÷ 8 = 11 or 88 ÷ 11 = 8)
• 11 × 5 (55, related division fact is 55 ÷ 5 = 11 or 55 ÷ 11 = 5)
• 12 × 8 (48, related division fact is 48 ÷ 8 = 12 or 48 ÷ 12 = 8)
• 11 × 3 (33, related division fact is 33 ÷ 3 = 11 or 33 ÷ 11 = 3)
• 12 × 5 (60, related division fact is 60 ÷ 5 = 12 or 60 ÷ 12 = 5)
• 12 × 7 (84, related division fact is 84 ÷ 7 = 12 or 84 ÷ 12 = 7)
• 11 × 9 (99, related division fact is 99 ÷ 9 = 11 or 99 ÷ 11 = 9)
• 12 × 3 (36, related division fact is 36 ÷ 3 = 12 or 36 ÷ 12 = 3)

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Maths is cool too! Some fun facts on mathematics

"Pure mathematics is, in its way, the poetry of logical ideas."- Albert Einstein

No matter how nice and simple this quote makes us think mathematics is, we are sure that for many of you, mathematics as a subject seems no less than a nightmare. The calculation which includes alphabets, let alone numbers, was no less than a herculean task to complete. However, the fact remains that Mathematics is an essential subject for students. Even though numbers can be scary sometimes, but if learned properly and with fun, they can be pretty amazing and cool.

To make our point, let's check out a few interesting facts about mathematics:

The lifeline of today's time, Google, derived its name from the word 'googol' -- a mathematical term for the number 1 followed by 100 zeros, which reflect infinite amount of search on the internet.

2. Crazy multiplications

A very interesting things about math is how crazy it gets with its function. For instance, if you multiply 111,111,111 by 111,111,111, this becomes equal to 12,345,678,987,654,321.

3. The terminology dilemma

This debate is going on for a long time now. But the answer depends on which part of the world you are in. Americans called mathematics 'math', saying that the function of the same is a singular noun and with that logic, they prefer saying 'math', which is singular too. On the other hand, speakers of British English would always say 'maths', as in 'I have a degree in maths'.

However, there are logical arguments for both the spellings. The Oxford and the Merriam-Webster dictionaries say the word is plural because of the letter 's' in the end. On the contrary however, it is usually used as a singular noun. For example, 'Mathematics is my favourite subject' and not 'Mathematics are my favourite subject'.

Other plural nouns that are used as if they were singular are economics, ethics, politics, gymnastics, measles and dominoes. These words, however, are not habitually shortened, making math/maths rather an unusual word.

Another mind-boggling application of maths comes in when the number 1 is divided by 998,001. The answer would give you a complete sequence from 000 to 999 in order. Don't agree with us? Go ahead and try it, and be ready to waste one entire notebook!

5. Pizza and math: Are they related?

We may seem like someone who is ruining pizza for you but you will be amazed to know its relation with maths. To find out the volume of the cylindrical shape of the pizza, the formula used is Pi x r 2 x h.

So, if an ordinary pizza has a radius of 'z' and height 'a', its volume is Pi x z x z x a which makes up 'pizza'.

6. Magical digit!

The number 9 is believed to be a magical number with certain very interesting properties. This is because if you multiply a number with 9, and add all the digits of the resulting number, the sum would always come out to be 9.

7. Zero is not there in Roman numerals

Did you know that one of the most important numbers, zero, is not represented in the Roman numerals? Derived from the Arabic word, 'sifr', it is known from a variety of other names like naught, zip, nil and zilch.

8. Did Shakespeare too have issues with maths?

No don't get us wrong, Shakespeare was a literature lover and not a math lover, but the only time he included the word 'mathematics' was in his play, The Taming of the Shrew.

9. What comes after a million?

After a million comes--- billion, trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion, and undecillion.

10. The growth of our mathematical knowledge

In 1900, all the world's mathematical knowledge could be written in 80 books today it would fill more than 1,00,000 books.

Here you will find a selection of Division sheets designed to help your child improve their understanding of what division is.

The sheets introduce the idea of division in terms of sharing and grouping, and designed to be a good practical start to learning about division.

All the free math work sheets in this section are informed by the Elementary Math Benchmarks for 2nd Grade.

Division Flashcards

Here you will find a selection of Division Flashcards designed to help your child learn their Division facts.

Using flashcards is a great way to learn your Math facts. They can be taken on a journey, played with in a game, or used in a spare five minutes daily until your child knows their facts off by heart.

All the free Math flash cards in this section are informed by the Elementary Math Benchmarks for 3rd Grade.

Multiplication & Division Times Table Charts

Here you will find a selection of Multiplication/Division Times Table Charts to 10x10 or 12x12 to support your child in learning their multiplication and division facts.

There is a wide selection of multiplication charts including both color and black and white, smaller charts, filled charts and blank charts.

Division Practice Area

Here is our free division practice area.

If you want to practice your division facts, or take a timed division test, then this is the place for you.

In this area, we cover the following division facts:

• division facts up to 5x5, up to 10x10 or up to 12x12
• division facts linked to individual tables facts
• dividing by 10 and 100.

Division Facts Worksheets

Here you will find a selection of Mental Division sheets designed to help your child improve their recall of Division Facts and to apply their facts to answer related questions.

Division (and multiplication) Worksheet Generator

Here is our free generator for division (and multiplication) worksheets.

This easy-to-use generator will create randomly generated division worksheets for you to use.

Each sheet comes complete with answers if required.

The areas the generator covers includes:

• Dividing with numbers to 5x5
• Dividing with numbers to 10x10
• Dividing with numbers to 12x12
• Divide with 10s e.g. 120 ÷ 4
• Divide with 100s e.g. 2100 ÷ 3
• Divide with tenths e.g. 2.4 ÷ 6
• Dividing with a single times table
• Practicing division with selected times tables

These generated sheets can be used in a number of ways to help your child with their division table learning.

Division Facts to 10x10 Sheets (3rd & 4th Grade)

Here you will find a selection of Division sheets designed to help your child learn their Division facts up to 10x10.

Example: if a child knows that 5 x 4 = 20, then they should also know that 20 ÷ 5 = 4 and 20 ÷ 4 = 5.

The sheets are graded so that the division facts start off up to 5x5, progressing on to 10x10 by the end.

• understand how division and multiplication are related
• learn their division facts up to 10x10.

Division Related Facts 10s and 100s (4th & 5th Grade)

Here you will find a selection of Division sheets designed to help your child learn to use their Division facts up to 10x10 to answer related questions.

Example: if you know that 42 ÷ 6 = 7, then you also know that 420 ÷ 6 = 70 or 420 ÷ 70 = 6, etc.

The sheets are graded so that the related division facts start off easier, then get gradually harder.

• know how to multiply and divide decimals up to 3dp by 10 or 100
• understand how division and multiplication are related
• apply their division facts up to 10x10 to answer related questions.

Division Related Facts Decimals (5th & 6th Grade)

Here you will find a selection of Division sheets designed to help your child learn to use their Division facts up to 10x10 to answer related questions involving decimals.

Example: if you know that 24 ÷ 6 = 4, then you also know that 2.4 ÷ 6 = 0.4 or 2.4 ÷ 0.4 = 6, etc.

• understand how division and multiplication are related
• apply their division facts up to 10x10 to answer related questions involving decimals.

We also have a generator for creating your own division worksheets involving signed integers.

You can choose the values you want and tailor the worksheets to your needs.

Long Division Sheets

Long Division starts properly once kids reach 3rd grade, and after they have a good understanding of what division is, and know their division facts.

Here you will find long division worksheets, starting from dividing a 2 digit number by a single digit, all the way up to dividing a 3 or 4 digit number by two digits.

Long Division 2 Digits by 1 Digit (3rd Grade)

Here you will find a selection of free Division sheets 3rd Grade which are designed to help your child understand how to do long division. The sheets are graded so that the easier ones are at the top.

Long Division 3 & 4 Digits by 1 Digit (4th Grade)

Here you will find a range of Long Division sheets which are designed to help your child master their Long Division by a single digit.

Long Division by 2 Digits (5th Grade)

Here you will find a selection of free Division sheets designed to help your child learn to do 2 digit long division. The sheets are graded so that the easier ones are at the top.

Long Division by Decimals (6th Grade)

We have some decimal division worksheets with up to 3 decimal places.

There are also some worked examples to show you how.

Division Word Problems

We have created lots of division word problems for you to solve.

The sheets involve solving division problems in a range of different contexts and involve both sharing and grouping.

These sheets involve solving a range of division problems.

Using this link will open our 2nd Grade Math Salamanders website in a new browser window.

Fun Division Games

Here you will find a range of Free Printable Division Games.

The following games develop the Math skill of dividing in a fun and motivating way.

All the printable Math sheets in this section are informed by the Elementary Math Benchmarks.

How to Print or Save these sheets

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Follow these 3 easy steps to get your worksheets printed out perfectly!

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Math-Salamanders.com

The Math Salamanders hope you enjoy using these free printable Math worksheets and all our other Math games and resources.

Division is most often shown by placing the dividend over the divisor with a horizontal line, also called a vinculum, between them. For example, a divided by b is written as

This can be read as "a divided by b", or "a over b". A way to express division all on one line is to write the dividend, then a slash, then the divisor, like this:

This is the usual way to specify division in most computer programming languages, since it can easily be typed as a simple sequence of characters.

A typographical variation which is halfway between these two forms uses a slash, but elevates the dividend and lowers the divisor:

Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (in which case, the two numbers are typically referred to as numerator and denominator). A fraction is an accepted way of writing numbers. It is not always expected that the result of the division is written in decimals.

A less common way to show division is to use the obelus (or division sign) in this manner:

But in elementary arithmetic this form is used rather often. The obelus is also used alone to represent the division operation itself, as in the case of a label on a key of a calculator.

In some non-English-speaking cultures, "a divided by b" is written as a : b. However, in English-speaking countries the colon is restricted to expressing the related concept of ratios Ώ] (where a:b reads "a is to b").

Long Division Teaching Aid, "Double Division"

Double Division does not depend on memorizing the multiplication facts or estimating how many times one number goes into another. It may take 50% longer, but it is far less frustrating and probably easier to understand than Long Division. example

Click "Step 1" on the calculator (above right) for step-by-step instructions.

Double Division may be easier to understand than Long Division because it deals with whole numbers rather than individual digits. You can see multiples of the divisor being subtracted from the dividend. In the example to the right, you see that 37 x 2,000 = 74,000 is being subtracted from the dividend of 85,434. (better notation?). With Long Division you'd be subtracting 74 from 85 neither of which are real numbers in the problem.

Extension to decimals (April 2008):
To extend the answer by one decimal place, start again with the remainder times ten. Repeat this as many times as you'd like, or until a remainder repeats itself. Decimal Examples

DOUBLE DIVISION DISCUSSIONS: (others?)

Video on Double Division at LearnZillion.com.

Writen procedure below, Bigger calculator here.

Long Division and Double Division side-by-side.

This method is a God send. This method sucks!

The value of teaching manual division is to give people a method they can actually use if they have to, and to teach about mathematics.

It could be argued that if students seldom use long division after leaving school then it might be better for them to have a simpler and more intuitive method at their disposal - one that is easier to remember and understand. Would someone ten years out of school have an easier time doing long division or Double Division? - I'm not sure, but Double Division seems simpler to me.

About teaching math, I'm not sure that long division teaches math very well. It helps teach people how to follow a long procedure and it gives some practice multiplying and subtracting. I'm not sure how many people understand what is happening when you bring down the next digit or understand why you have to add a zero to the answer when the "number after subtracting" is less than the divisor.

One problem I see with Double Division is that you have to do more subtraction. I personally find subtraction harder than multiplication. The other problem is that (not counting the trial and error) Double Division will usually have more steps.

1) In long division you guess which multiple to subtract, where as in Double Division you pick from four options. (Any multiple you pick works. You don't always have to pick the largest possible one sometimes I pick the easiest one to subtract.)

2) In Double Division you write out the zeros so it's clear how big the numbers are.

3) In Double Division you write your answer on the right side where you have room to accumulate many parts of the answer. In long division you write the answer on the top and have to get each digit of the answer exactly right - because there is only room there to write one number.

Here is how I see it so far:

REASONS TO TEACH DIVISION:

- a method to actually use in rare instances

- to prepare students for higher math

- teaches how division works

- no trial and error: I don't know how many times 372 go into 2711.

- easier to do: Double, double, double, subtract the best ones, add up the answer.

- gives practice doubling numbers

- more steps (see note below)

- requires more space on the paper

- incremental difficulty in going on to decimals ?may? be more

- may not lead as directly to polynomial division - arguable

IS Double DIVISION LONGER?

If we assume there is an equal chance of all ten digits being in the answer then on average there will be 1.5X as many "multiple and subtract" steps. For example a "7" in the answer requires 3 steps, and a zero in the answer requires no steps.

Also remember that the multiply part of the "multiply and subtract" steps is already done for you. So this part will be faster. Of course you have to pre-multiple the divisor three times in the beginning.

In the end I think it is longer, but not as much as you might think initially.

FEEDBACK SO FAR - 10/14/2005

The response has been mixed. A few people have thought doubling the divisor three times was an efficient way to get trial multiples, and liked the idea of not having to guess. Several people have said this method is no easier than long division. I think the challenge is comparing the "easiness" of one thing that you know to another thing you are just learning. But on the other hand, this explanation of long division is pretty simple.

On October 11, 2005 a sixth grade teacher posted this comment after teaching Double Division to his class.