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:

  1. 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

  1. 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.

45 ÷ 9 = 5
45 – 9 = 36 1 time
36 – 9 = 27 2 times
27 – 9 = 18 3 times
18 – 9 = 9 4 times
9 – 9 = 0 5 times / with a remainder:
33 ÷ 6 = 5 ½
33- 6 = 27 1 time
27 – 6 = 21 2 times
21 – 6 = 15 3 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…

Zero CAN BE a DIVIDEND, but the quotient/answer will be 0 (thumbs up- you can do it).

0 ÷ 35 = 0

3 Ways to Write Divison:

  1. Division Sentence
  1. Fraction
  1. Using a Math Symbol

Sentence / Fraction / Math Symbol
56 ÷ 4

Estimating Quotients


192 ÷ 9
180 ÷ 9 = 20
412 ÷ 6 Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48
420 ÷ 6 = 70
or
360 ÷ 6 = 60

264 ÷ 32
300 ÷ 30 = 10

Remainders in Division

(Use the 4 step problem solving method)

3 options:

  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

  1. 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.

  1. 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.**

  1. 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.

  1. 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 x = y ÷ 25
  3. show work:

84 + 12 = 96

96 ÷ 25 = 3 21/25

4. x = (84 + 12) ÷ 25 = 3 21/25 = 4 buses are needed.

You round the remainder up so all the people can have a seat for the trip.

Remainders in Division

Drop/Ignore it / Keep it / Round up
  • “full” or “whole”
  • You have some thing/object you can’t break into fractional parts
  • Remainder is not needed in your final answer.
/
  • Usually deals with money $$
  • You can break whatever it is into fractional parts.
  • Exact amount is given- no more, no less.
/
  • 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.

Examples:
Mrs. C put books into boxes. She had 32 books. Each box held 5 books. How many full boxes did she pack?
Jane had 42 dollars to spend on CDs. Each CD cost $11. How many CDs did Jane buy? / Example:
Seamus spent $9 on dog bones. He bought 6 bones. How much did each bone cost? / Example:
90 people went on a bus to the park. Each bus held 25 people. How many buses did they need to order?

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”

  • “how many are in each…”
  • “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.