B. Number · Operations: Rational Numbers, Integers, Fractions and Proportional Reasoning

Dividing Integers and the Order of Operations with Integers

B2.7 B2.1 A1About 45 minutes

Learning goals

I can divide integers and explain the sign of the quotient using multiplication.
I can find the mean of a set of integers, like temperatures.
I can use the order of operations with integers, including powers with negative bases.
I can check a quotient by multiplying.

Words to know

quotient
The answer to a division.

dividend
The number being divided. In (−20) ÷ 4, the dividend is −20.

divisor
The number you divide by. In (−20) ÷ 4, the divisor is 4.

mean
The total of a set of numbers divided by how many numbers there are.

undefined
Has no value. Dividing by zero is undefined.

Remember

In the last lesson you learned to multiply integers: same signs give a positive product and different signs give a negative product. Division undoes multiplication, so the same sign rules work for division.

Let's start with an evening temperature drop

Between 4 p.m. and 10 p.m. (6 hours), the temperature fell 18 degrees. What was the average change per hour?

Share the change of −18 equally over 6 hours: (−18) ÷ 6 = −3. The temperature went down about 3 degrees each hour.

Check with multiplication: 6 × (−3) = −18. ✓

The total change of −18 split into 6 equal hourly jumps gives −3 each hour: (−18) ÷ 6 = −3. The labels show the change since 4 p.m., not the actual temperature.

Multiplication fact	Division facts from it
4 × (−5) = −20	(−20) ÷ 4 = −5 and (−20) ÷ (−5) = 4
(−6) × (−7) = +42	42 ÷ (−6) = −7 and 42 ÷ (−7) = −6
(−3) × 8 = −24	(−24) ÷ 8 = −3 and (−24) ÷ (−3) = 8

Every integer multiplication fact gives two division facts.

A fact family links one multiplication fact to two division facts. Cover any circle and use the other two to find it: (−20) ÷ (−5) = 4.

The sign rules for division

Same signs → positive quotient: (−20) ÷ (−5) = +4.
Different signs → negative quotient: (−20) ÷ 4 = −5, and 42 ÷ (−6) = −7.
0 divided by any integer (except 0) is 0: 0 ÷ (−7) = 0.
Dividing by 0 is undefined. No number times 0 gives −5, so (−5) ÷ 0 has no answer.

Division is often written as a fraction: −12/4 = −3. In fact, (−12)/4, 12/(−4) and −(12/4) all equal −3.

The order of operations still applies

With integers, use BEDMAS exactly as before. Put brackets around negative numbers so you can see which minus signs are operations and which are signs.

Worked example

Example 1: Same signs

Find (−48) ÷ (−6).

Same signs → positive.
48 ÷ 6 = 8.
Check: (+8) × (−6) = −48. ✓

Worked example

Example 2: Different signs

Find (+63) ÷ (−9).

Different signs → negative.
63 ÷ 9 = 7.
Check: (−7) × (−9) = +63. ✓

−7

Worked example

Example 3: Mean temperature

The low temperatures for five days in January were −5°C, −8°C, +2°C, −3°C and −6°C. What was the mean low temperature?

Add: negatives (−5) + (−8) + (−3) + (−6) = −22. Then (−22) + (+2) = −20.
Divide by the number of days: (−20) ÷ 5 = −4.

−4°C

Worked example

Example 4: Order of operations

Evaluate (−3) × 4 − (−20) ÷ (−5) + (−2)².

Exponent: (−2)² = +4.
Multiply and divide, left to right: (−3) × 4 = −12, and (−20) ÷ (−5) = +4.
Now (−12) − (+4) + (+4). Add the opposite: (−12) + (−4) + (+4) = −12.

−12

Worked example

Example 5: Brackets first

Evaluate [(−18) + 6] ÷ (−3).

Brackets: (−18) + 6 = −12.
(−12) ÷ (−3) = +4.

Check by multiplying

Every division can be checked with multiplication: quotient × divisor should equal the dividend. If (−56) ÷ (−8) = 7, then 7 × (−8) should be −56. It is. ✓

Hands-on activity

Fact family cards

You need: a deck of cards with the 10s and face cards removed (Ace = 1), paper, a pencil. Black cards are positive and red cards are negative.

Draw two cards and multiply them.
Write the two division facts that come from your multiplication fact.
Draw five more cards and find their mean: add them, then divide by 5. If the mean is not a whole number, write it as a decimal.
Repeat a few times. Check each division by multiplying.

Model answer: red 6 and black 4 → (−6) × (+4) = −24, so (−24) ÷ (+4) = −6 and (−24) ÷ (−6) = +4. Five cards red 3, red 7, black 2, red 1, black 4 → sum (−11) + (+6) = −5, and the mean is (−5) ÷ 5 = −1.

Just for fun

Ask a calculator to divide by 0 and it just shows Error. Even a machine that can do millions of calculations a second knows some questions have no answer.

Your turn

(−36) ÷ (+4) = ?

(−56) ÷ (−8) = ?

(+45) ÷ (−5) = ?

0 ÷ (−7) = ?

(−2) × (−6) + (−15) = ?

(−40) ÷ [(−3) + (−5)] = ?

A diver goes from the surface to −36 m in 4 minutes at a steady rate. What is her change in depth per minute?

More practice: warm-up

(−10) ÷ (+2) = ?

(−21) ÷ (−3) = ?

(+18) ÷ (−6) = ?

(−9) ÷ (−9) = ?

(+30) ÷ (+5) = ?

More practice: on your own

(−72) ÷ (−9) × (−2) = ?

(−5)² − 4 × (−3) = ?

[(−6) − 10] ÷ (−4) = ?

Find the mean of −12, −7, +4 and −1.

Find the missing number: ___ ÷ (−6) = 7.

Challenge

Evaluate (−3)³ ÷ 9 − (−8) ÷ 2².

Over 5 days, the water level of a pond changed by +3 cm, −8 cm, −6 cm, +1 cm and one more amount. The mean change was −3 cm per day. What was the fifth change?

Which is greater, (−100) ÷ (−4) or (−100) ÷ 4? By how much?

Check your understanding

Quiz

What is (−42) ÷ 7?

What is (−50) ÷ (−10)?

What is 4 + (−12) ÷ (−3)?

What is the mean of −6, −2 and +2?

Which of these is undefined?

What is (−2)² × (−3)?

For parents and teachers

Big idea: This lesson finishes B2.7 (multiply and divide integers) and connects it to B2.1 (order of operations with rational numbers). Integers are rational numbers, so everything from the order-of-operations lesson still applies.

Common difficulties:

Treating every minus sign the same. In 25 − (−12), the first minus is subtraction and the second is the sign of −12. Brackets help.
Doing operations left to right without BEDMAS, or doing all multiplication before division.
Means of integers: forgetting that a negative total gives a negative mean.

How to help: Look up the forecast lows for the next five days and have your child find the mean. Ask them to check every division with a multiplication.