Number — Year 7
Working with Integers
Concept
What even is a negative number?
You already know positive numbers — your everyday counting numbers.
Negative numbers are what happen when you go the other way.
Think temperature: 5°C is cold. −5°C is colder. −20°C is "why would anyone live here" cold.
Or think money:
💰 +$10 = you have ten dollars
💸 −$10 = you OWE ten dollars
Negative numbers aren't weird — they're everywhere. Once you can work with them, a whole lot of maths unlocks.
Negative numbers are what happen when you go the other way.
Think temperature: 5°C is cold. −5°C is colder. −20°C is "why would anyone live here" cold.
Or think money:
💰 +$10 = you have ten dollars
💸 −$10 = you OWE ten dollars
Negative numbers aren't weird — they're everywhere. Once you can work with them, a whole lot of maths unlocks.
L1 — Adding and subtracting integers
Adding a positive → move RIGHT. You get more.
5 + 3 = 8 ✓
Adding a negative → move LEFT. You get less.
5 + (−3) = 2
Think of it as: someone dumps a $3 debt into your account. You go from $5 to $2.
Shortcut: + (−) becomes just −
5 + (−3) = 5 − 3 = 2
5 + 3 = 8 ✓
Adding a negative → move LEFT. You get less.
5 + (−3) = 2
Think of it as: someone dumps a $3 debt into your account. You go from $5 to $2.
Shortcut: + (−) becomes just −
5 + (−3) = 5 − 3 = 2
Subtracting a positive → move LEFT. Normal.
8 − 3 = 5 ✓
Subtracting a negative → move RIGHT. This is the tricky one!
5 − (−3) = 8
Why? Removing a debt makes you richer. If someone cancels your $3 debt, you're $3 better off.
Shortcut: − (−) becomes +
5 − (−3) = 5 + 3 = 8
8 − 3 = 5 ✓
Subtracting a negative → move RIGHT. This is the tricky one!
5 − (−3) = 8
Why? Removing a debt makes you richer. If someone cancels your $3 debt, you're $3 better off.
Shortcut: − (−) becomes +
5 − (−3) = 5 + 3 = 8
When two signs sit next to each other, simplify first:
+ + → +
+ − → −
− + → −
− − → +
Same signs = positive. Different signs = negative.
Examples:
7 − (−4) → 7 + 4 = 11
−3 + (−5) → −3 − 5 = −8
Always simplify signs FIRST, then calculate.
+ + → +
+ − → −
− + → −
− − → +
Same signs = positive. Different signs = negative.
Examples:
7 − (−4) → 7 + 4 = 11
−3 + (−5) → −3 − 5 = −8
Always simplify signs FIRST, then calculate.
Step by step
Worked example — 5 + (−3)
We have + followed by (−). Two signs next to each other.
+ and − → −
Rewrite: 5 + (−3) becomes 5 − 3
+ and − → −
Rewrite: 5 + (−3) becomes 5 − 3
5 − 3 = 2
Straightforward subtraction once signs are cleaned up.
Straightforward subtraction once signs are cleaned up.
We added a negative, so the result should be LESS than 5.
2 < 5 ✓ Makes sense!
2 < 5 ✓ Makes sense!
5 + (−3) = 2
Step by step
Worked example — −15 − (−20) + 4
We have − (−20). Same signs → positive.
−15 − (−20) + 4 becomes −15 + 20 + 4
−15 − (−20) + 4 becomes −15 + 20 + 4
−15 + 20 = 5
Jumped from −15, moved right 20 steps, landed on 5.
Jumped from −15, moved right 20 steps, landed on 5.
5 + 4 = 9
−15 − (−20) + 4 = 9
Concept
L2 — Multiplying and dividing integers
Here's the big idea: when you multiply or divide, you only need to worry about TWO things separately.
1. What is the answer if I ignore the signs? (just multiply/divide the numbers)
2. What sign does the answer get? (use the sign rules below)
Do both steps and you're done.
1. What is the answer if I ignore the signs? (just multiply/divide the numbers)
2. What sign does the answer get? (use the sign rules below)
Do both steps and you're done.
Sign rules — quick reference
(+) × (+) = +
3 × 4 = 12
Same signs → positive
Same signs → positive
(−) × (−) = +
−3 × −4 = 12
Same signs → positive
Same signs → positive
(+) × (−) = −
3 × −4 = −12
Different signs → negative
Different signs → negative
(−) × (+) = −
−3 × 4 = −12
Different signs → negative
Different signs → negative
Step by step
Worked example — −4 × 5
−4 is negative. 5 is positive.
Different signs → result will be NEGATIVE.
Different signs → result will be NEGATIVE.
4 × 5 = 20
Result is negative → −20
−4 × 5 = −20
Step by step
Worked example — (−12 × −4) ÷ (−2)
Calculate −12 × −4.
Same signs (both negative) → positive.
12 × 4 = 48, so −12 × −4 = +48
Same signs (both negative) → positive.
12 × 4 = 48, so −12 × −4 = +48
48 ÷ (−2)
Different signs → negative.
48 ÷ 2 = 24, so 48 ÷ (−2) = −24
Different signs → negative.
48 ÷ 2 = 24, so 48 ÷ (−2) = −24
(−12 × −4) ÷ (−2) = −24
Flip cards — tap to reveal
Test yourself — sign rules
(+) × (+) = ?
Positive ✓
Same signs → positive
Same signs → positive
(−) × (−) = ?
Positive ✓
Two negatives → positive
Two negatives → positive
(+) × (−) = ?
Negative ✗
Different signs → negative
Different signs → negative
(−) ÷ (−) = ?
Positive ✓
Same signs → positive
Same signs → positive
−(−5) = ?
+5
Subtracting a negative = adding a positive
Subtracting a negative = adding a positive
Is (−3)² positive or negative?
Positive!
(−3) × (−3) = +9
Same signs → positive
(−3) × (−3) = +9
Same signs → positive
Note
⚠️ Classic trap — −2² vs (−2)²
These look similar. They are NOT the same.
−2² = −4
Square the 2 first (= 4), then negate. The negative sits outside the power.
(−2)² = +4
The bracket means square the whole thing: (−2) × (−2) = +4.
Same digits. Very different answers. Read the brackets!
−2² = −4
Square the 2 first (= 4), then negate. The negative sits outside the power.
(−2)² = +4
The bracket means square the whole thing: (−2) × (−2) = +4.
Same digits. Very different answers. Read the brackets!
Step by step
Hard example — −96 ÷ 12 ÷ (−2)²
(−2)² = (−2) × (−2) = +4
Same signs → positive.
Same signs → positive.
−96 ÷ 12 ÷ 4
Now it's a clean left-to-right division chain.
Now it's a clean left-to-right division chain.
−96 ÷ 12 = −8
Different signs → negative.
Different signs → negative.
−8 ÷ 4 = −2
Different signs → negative.
Different signs → negative.
−96 ÷ 12 ÷ (−2)² = −2
Step by step
Real world — The diver problem
A diver is 12 m below sea level and descends 5 m more. What is the new depth?
Below sea level = negative. Starting position = −12 m.
Descending further = going more negative.
New depth = −12 − 5
Descending further = going more negative.
New depth = −12 − 5
−12 − 5 = −17
Moved 5 more steps in the negative direction.
Moved 5 more steps in the negative direction.
New depth = −17 m (17 m below sea level)
Practice
Your turn — quick check
1Easy: Calculate 5 + (−3)
Simplify: +(−) → −
5 + (−3) = 5 − 3 = 2
5 + (−3) = 5 − 3 = 2
2Easy: Calculate −4 × 5
Different signs → negative
4 × 5 = 20
Answer: −20
4 × 5 = 20
Answer: −20
3Easy: What is −(−8)?
−(−) → +
−(−8) = +8 = 8
−(−8) = +8 = 8
4Intermediate: Calculate −15 − (−20) + 4
Simplify: −(−20) → +20
−15 + 20 + 4 = 5 + 4 = 9
−15 + 20 + 4 = 5 + 4 = 9
5Intermediate: Calculate (−12 × −4) ÷ (−2)
Brackets: −12 × −4 = +48 (same signs)
Then: 48 ÷ (−2) = −24 (different signs)
Answer: −24
Then: 48 ÷ (−2) = −24 (different signs)
Answer: −24
6Hard: Evaluate −96 ÷ 12 ÷ (−2)²
Orders: (−2)² = +4
−96 ÷ 12 = −8
−8 ÷ 4 = −2
Answer: −2
−96 ÷ 12 = −8
−8 ÷ 4 = −2
Answer: −2
7Real world: A diver is 12 m below sea level and descends 5 m more. Find the new depth.
Start: −12 m
Change: −12 − 5 = −17 m
Answer: 17 m below sea level (−17 m)
Change: −12 − 5 = −17 m
Answer: 17 m below sea level (−17 m)