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Dividing Whole

I. Understanding :

1. Define division______

______

2. Show how many groups of 6 apples there are in 18 apples. (Circle each 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.

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 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 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