Division Definitions
Dividend: The total number that you start with before fair sharing or making
equal groups.
Divisor: The number of equal groups.
Quotient: The answer to a division problem.
Remainder: The amount leftover after “fair sharing” or “making equal groups”.
Divisibility: A number “a” is divisible by a number “b” if “b” divides evenly
into “a” with no remainder.
Division Unit
Division means:
I. Fair Share – giving out or sharing ONE AT A TIME.
*Playing Cards – hand them out one at a time
25 ÷ 5 = 5
25 cards are divided among 5 people- each person gets 5 cards.
lllll lllll lllll lllll lllll
II. Make Equal Groups – give out or share THE ENTIRE
AMOUNT at a time.
50 ÷ 10 = 5
10
10
10
10
10
50 chips, put into baggies of 10 are handed out, a group at a time,
to 5 people.
** Division and Multiplication are INVERSE OPERATIONS**
DIVISION is REPEATED SUBTRACTION.
with a remainder:
45 ÷ 9 = 5
33 ÷ 6 = 5 ½
45 – 9 = 36 1 time
36 – 9 = 27 2 times
33- 6 = 27 1 time
27 – 9 = 18 3 times
27 – 6 = 21 2 times
18 – 9 = 9
4 times
21 – 6 = 15 3 times
9–9=0
5 times
15 – 6 = 9
4 times
9–6=3
5 times 3/6
Division with Zero
Zero can NEVER be a DIVISOR (thumbs down – Cannot do it).
Proof: 35 ÷ 0 – remember that division is repeated subtraction.
35 – 0 = 35
35 – 0 = 35
35 – 0 = 35…
35 is unchanged, making
division by zero
meaningless or
UNDEFINED.
Zero CAN BE a DIVIDEND, but the quotient/answer will be 0 (thumbs upyou can do it).
0 ÷ 35 = 0
3 Ways to Write Divison:
I. Division Sentence
II. Fraction
III. Using a Math Symbol
Sentence
56 ÷ 4
Fraction
Math Symbol
Estimating Quotients
192 ÷ 9
180 ÷ 9 = 20
Think of MULTIPLES of the
divisor (9) and round the
dividend to that multiple. All
remaining digits become zero.
412 ÷ 6
420 ÷ 6 = 70
or
360 ÷ 6 = 60
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48
264 ÷ 32
1. ROUND the DIVISOR
300 ÷ 30 = 10
2. Find the CLOSEST
MULTIPLE to the new
divisor.
3 options:
Remainders in Division
(Use the 4 step problem solving method)
1. Ignore it/Drop it
-
the remainder is NOT NECESSARY in the answer OR it can’t be
used in the answer.
Example: Sarah has $50 to buy CDs. Each CD costs $7. How many CDs can
Sarah buy?
1. x = Number of CDs Sarah bought.
2. X = 50 ÷7
3. show work:
50 ÷ 7 = 7 1/7
4. x = 50 ÷7 = 7 1/7 CDs = 7 CDS
You drop the remainder because you cannot buy 1/7 of a CD OR you
cannot afford 8 CDs.
2. Keep the remainder
-
the remainder is necessary for your answer to be correct.
Example: Sean spent $60 on candy. He bought 8 boxes to give as gifts. How
much did each box of candy cost?
1. x = Cost per box of candy.
2. X = 60 ÷8
3. show work:
60 ÷ 8 = 7 ½ = $7.50
** When working with money, make sure you turn the remaining fraction into
cents.**
4. x = 60 ÷8 = $7.50 per each box of candy.
The remainder must be kept in the answer because it represents the
exact amount of each box.
3. Round the remainder
-
the fractional part that is leftover must be rounded up to a whole
number.
Example: The 5th grade students are going on a trip to Midieval Times.
Buses need to be reserved for 84 students and 12 adults. Each bus holds 25
people. How many buses will be needed?
1. x = Number of buses needed
2. X = (84 + 12) ÷ 25
OR y = 84 +12
3. show work:
84 + 12 = 96
96 ÷ 25 = 3 21/25
4. x = (84 + 12) ÷ 25 = 3 21/25 = 4 buses are needed.
x = y ÷ 25
You round the remainder up so all the people can have a seat for the
trip.
Remainders in Division
Drop/Ignore it



“full” or “whole”
You have some
thing/object you
can’t break into
fractional parts
Remainder is not
needed in your final
answer.
Examples:
Mrs. C put books into
boxes. She had 32 books.
Each box held 5 books.
How many full boxes did
she pack?
Keep it



Round up
Usually deals with
money $$
You can break
whatever it is into
fractional parts.
Exact amount is
given- no more, no
less.
Example:
Seamus spent $9 on dog
bones. He bought 6 bones.
How much did each bone
cost?


You have some
thing /object that
you can’t break into
fractional parts
You need the
remaining parts
(numerator in the
fraction) in your
answer.
Example:
90 people went on a bus to
the park. Each bus held 25
people. How many buses
did they need to order?
Jane had 42 dollars to
spend on CDs. Each CD
cost $11. How many CDs
did Jane buy?
 The dividend and the quotient have a relationship. The numerator in
the quotient is whatever is left over after the dividend has been put
into equal groups.
 Key word for division and multiplication: “each”
o “how many are in each…”
o “if each…”
Dividing with Multiples of 10
1. Divide the number fact you know.
2. math up the zeroes – “dance partners”
3. the remaining zeroes are written in the answer.
72,000 ÷ 800 = 90
Check: 800 x 90 = 72,000
4,200,000 ÷ 60 = 70,000
Check: 60 x 70,000 = 4,200,000
12,100 ÷ 110 = 110
Check: 110 x 110 = 12,100
56, 000 ÷ 80 = 700.
Check: 700 x 80 = 56,000
400,000 ÷ 50 = 8,000
Check: 8,000 x 50 = 400,000
Short Cut Division
 Use when you have ONE digit divisor ONLY; the size of the
dividend does not matter.
 The process where you divide and record the remainders
horizontally.
 Multiplication and subtraction steps are done mentally. The
remainder is written as a subscript and placed before the
next digit.