DIVISION PRIMER |

**What is Division?**

Multiplication is repeated addition. **Division** is the **inverse** operation (opposite action) **of multiplication**, so -- **division is** **repeated subtraction**. If **4** of us **share** the cost of a **$12** pizza, we **each pay $3** because 4 × 3 = 12, therefore 12 ÷ 4 = 3. This means we could separate $**12** into **4 "equal packets"** of **$3** each.

12 - 3 - 3 - 3 - 3 = 0

But since **division is the inverse operation of multiplication**, and repeated subtraction could become extremely time consuming once the numbers are big -- we **use** the **multiplication** **tables**, combined with our knowledge of **factors** and **estimating**, **to** help us **divide** numbers efficiently.

**Fractions mean division**. When we **take ½** of anything, we **divide** it **by 2**. When we **find ¼,** we **divide by 4**. And, since **¾ is the same as 3 × ¼** , when we take** ¾** of something, we **divide it by 4** and **multiply it by 3**.

**Division Vocabulary**

Since math is a language, we have to pay special attention to the words we use to name the different numbers and symbols in our work. We use 3 terms -- multiplier, multiplicand and product -- to name the parts of a multiplication expression. We use **4 terms to name** the parts of a **division** expression.

The **dividend** is the **number** **to** **be divided**. If the division is presented as **a fraction, **it is** the numerator**. The **number** doing the **dividing** is called **the divisor**, or **the denominator** (in the fraction). The **result** or answer is called the **quotient**. At times, there is a **remainder** -- which, like its name says -- is **left over** once we subtract all the whole-number "** divisor-sized packets**" we can, from the dividend. When we divide an odd number by an even one, there will always be a remainder of 1, since any odd number is 1 bigger than the preceding even number.

Say 3 of us want to share 7 candy bars ** equally**. After each of us takes 2 whole candy bars, there will be

If our pizza had cost $13 instead of $12, an equal share of the cost would be **13 ÷ 4 = 3¼ or 3.25**. Once **we **each** pay $3**, there will be **$1-more left to pay**. So, each of us contributes another $¼ or 25¢, and the 13th dollar is paid.

In the 2nd fraction above -- the one with **15 + 2** in the numerator.

We **rewrote** or regrouped **17** as **15 + 2** because **15** = 5 × 3 -- **a multiple of 5** and therefore divisible by 5 -- and since we're trying to divide by 5 -- that's a good thing!! 15 is also the **greatest whole number multiple of 5** that **we can **__subtract__** from 17** -- and that's what we want. When we **divide** **15 by 5, **we** get 3** -- the whole number part of the quotient. We have a **remainder of 2** that needs to be **divided by 5** -- so our answer is . Notice the remainder of 2 on the right side.

If 5 of us shared a cab that cost $17, we'd each have to contribute $3.40

because two-fifths of a dollar = 40¢.

**what to do with the remainder?**

There are 2 ways to deal with the remainder. Either we express the quotient as a mixed number with fraction part made up of the remainder over the divisor, or we write **" R "** and the remainder value next to the quotient -- shown above in the example on the right.

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**Examples with no Remainder:**

Here is the 7 times table. It lists **multiples of 7** from 7 × 1 to 7 × 9.

× |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |

7 |
7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 |

It tells us that **14 ÷ 7 = 2**, and **42 ÷ 7 = 6** and so on.

If we divide any number in the 2nd row by 7 we will get a whole number quotient = to the number in the 1st row. So when we divide a multiple of 7 by 7, there will be no remainder.

**Examples with Remainders:**

When we divide a number that is not a multiple of 7 by 7, there will be a remainder.

**Checking our Division**

When we do math questions, we must always check our work, because mathland is a perfect place where coming close is no good at all. When we do math, we aim for perfection.

**To check** a **division** question:

**multiply quotient × divisor** then **add** the **remainder**.

In the examples above:

(5 × 7) + 4 = 39, | (3 × 7) + 2 = 23 | (7 × 7) + 5 = 54 |

**Summary of Steps in Division**

1. For single-digit divisors, **divide**, **multiply** and **subtract**.

2. Express **remainders** as **fractions** or with uppercase **" R " **and the remainder's value.

4. To check, **multiply** **quotient by divisor**, **add** the **remainder**, **get** the **dividend**.

Now get a pencil, an eraser and a note book, copy the questions,

do the practice exercise(s), then check your work with the solutions.

If you get stuck, review the examples in the lesson, then try again.

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**Practice Exercises**

1) Use the multiplication tables to find these quotients.

a) 56 ÷ 8 = | b) 25 ÷ 5 = | c) 45 ÷ 9 = | d) 32 ÷ 4 = |

e) 63 ÷ 7 = | f) 24 ÷ 3 = | g) 48 ÷ 6 = | h) 81 ÷ 9 = |

2) Find these quotients and remainders.

a) 59 ÷ 8 = | b) 26 ÷ 5 = | c) 47 ÷ 9 = | d) 35 ÷ 4 = |

e) 69 ÷ 7 = | f) 23 ÷ 3 = | g) 51 ÷ 6 = | h) 88 ÷ 9 = |

3) Shawna, Jennifer, Tanya and Sofia shared a taxi when they went to the museum. The meter said **$12.50** and they wanted to tip the driver **$1.50**. What was each girl's share of the taxi fare?

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**Solutions**

1) Use the multiplication tables to find these quotients.

a) 56 ÷ 8 = 7 |
b) 25 ÷ 5 = 5 |
c) 45 ÷ 9 = 5 |
d) 32 ÷ 4 = 8 |

e) 63 ÷ 7 = 9 |
f) 24 ÷ 3 = 8 |
g) 48 ÷ 6 = 8 |
h) 81 ÷ 9 = 9 |

2) Find these quotients and remainders.

a) 59 ÷ 8 = 7 R3 |
b) 26 ÷ 5 = 5 R1 |
c) 47 ÷ 9 = 5 R2 |
d) 35 ÷ 4 = 8 R3 |

e) 69 ÷ 7 = 9 R6 |
f) 23 ÷ 3 = 7 R2 |
g) 51 ÷ 6 = 8 R3 |
h) 88 ÷ 9 = 9 R7 |

3) The meter plus the tip = $14, then we ÷ 4 to get 3 R2 -- and since 2/4 = ½,

each girl's share of the taxi fare is **$3.50**.

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