Dividing Whole Numbers
I. Understanding Division:
1. Define division______
______
2. Show how many groups of 6 apples there are in 18 apples. (Circle each group of 6 apples; how many groups are there?______
3. There are three ways to write "18 divided by 6" or "6 divided into 18" 3 6)18 18 ÷ 6 = 3 18 = 3 6 Know the names of the parts of a division problem written in these forms.
quotient dividend ÷ divisor = quotient divisor )dividend
dividend = quotient divisor
Write the problems and find the quotient. Show all three ways to write this. The dividend is 20 and the divisor is 4.
÷ = =
This instructional aid was prepared by the Tallahassee Community College Learning Commons. 4. Division and Multiplication are Related Operations
10 ÷ 2 = 5 because 5 2 × 5 = 10 2)10 10 = 5 divisor × quotient = dividend 2
(It's easy to check division by multiplying!)
32 a. 6 ÷ 2 = 3 b. 4 40 c. = ______4
____ × _____ = ______× _____ = ______× _____ = _____
5. Special Division Problems:
6 a. 6 ÷ 6 = 1 6 6 = 1 6
A number divided by itself is 1. (There is one exception. Zero divided by itself is NOT 1.)
7 b. 7 ÷ 1 = 7 1 7 = 7 1 A number divided by one is the same number.
Before we consider zero and division you must understand: for an operation to be possible (or defined), there must be one and only one answer.
6. Consider the problem "Zero divided by seven."
0 7 0 0 ÷ 7 7 Think of the related multiplication:
divisor 6 6 quotient = dividend
7 × _____ = 0
What number makes this true?______
2 Is it the only number that makes it true? (Yes.)
7. Consider the problem "five divided by zero." 5 5 0 5 ÷ 0 0 Think of the related multiplication:
divisor × quotient = dividend
0 × _____ = 5
What number makes this true?______Since there is no answer, it is impossible to divide zero into another whole number.
8. Consider the problem "zero divided by zero." 0 0 0 0 ÷ 0 0 Think of the related multiplication:
divisor × quotient = dividend
0 × _____ = 0
What number makes this true?______Is that the only number that makes it true? (No, because 0 × any number = 0) Since there is more than one answer, it is impossible to divide zero by itself.
Looking at 6, 7 and 8, you can see it's impossible to divide by zero. Zero can be a dividend, but zero can never be a divisor.
0 5 0 = 0 is undefined is undefined 3 0 0 (or impossible)
3 because 3 • 0 = 0 0 • ___ = 5 There are too many, has no answer too many “answers” to 0 • _____ = 0
9. The Commutative and Associative Properties do NOT apply to division.
6 a. 2 6 , , 6 ÷ 2 ask 2
"How many groups of 2 are in 6?"
______The answer is 3. 2 6)2 , 6 2 ÷ 6 ask
"How many groups of 6 are in 2?"
There is less than one group (if 6 makes a group).
*Changing the order of the numbers to be divided changes the answer. You must know which number is the divisor and which is the dividend. (This is important when you use a calculator, too.)
b. Study these problems. REMEMBER the operation inside the parentheses is done first.
1. (36 ÷ 6) ÷ 2 2. 36 ÷ (6 ÷ 2) 6 ÷ 2 36 ÷ 3 3 12
Changing the grouping in a division problem changes the answer when there are NO parentheses. *When there are NO parentheses and there is more than one division in a problem, we have the agreement to work from left to right (the way we read)!
4 II. PROBLEMS: 0 7 1. 100 ÷ 20 ÷ 5 2. 3. 9 0
18 0 9 4. 5. 6. 18 0 1
15 7. 24 ÷ (12 ÷ 2) 8. 3
9. Use #8: The quotient is______The divisor is ______The dividend is______
ANSWERS:
I. 1. Division is used to separate objects into equal groups. 2. δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ
5 20 3. 4 20 20 ÷ 4 = 5 = 5 4 4. a. 2 × 3 = 6 b. 4 × 10 = 40 c. 4 × 8 = 32
6. 7 × 0 = 0
7. There is no answer because 0 × any number = 0 The product will never be 5!
8. 0 × 0 = 0 Any number you multiply by 0 will have a 0 × 1 = 0 product of 0 0 × 2 = 0
5 II. 1. 100 ÷ 20 ÷ 5 2. 0 3. Undefined or 5 ÷ 5 impossible 1 (never divide by 0)
4. 1 5. Undefined or 6. 9 impossible (never divide by 0)
7. 24 ÷ 6 8. 5 4
9. quotient is 5, divisor is 3, dividend is 15
(this is also true of 3 15 and 15 ÷ 3 = 5)
This instructional aid was prepared by the Tallahassee Community College Learning Commons. 6