Lesson Guide
Explanation
Negative numbers aren't strange — you already use them. A freezing morning is −5°C, a bank account can be $20 in the red, a diver swims 40 m below sea level. On the number line, negatives sit to the left of zero, and left always means smaller. Every number has a mirror-image opposite on the other side of zero: 3 and −3.
Your Turn
Think of one more real-life negative (lift levels? golf scores? goal difference?). Then decide: which is smaller, −8 or −2? Check: −8 — think temperature: −8°C is colder.
Watch Out
It's tempting to say −8 is bigger than −2 because 8 > 2. Anchor it to temperature: −8°C is colder, so −8 is smaller.
Explanation
Adding is walking along the line. Adding a positive walks right; adding a negative walks left. For 3 + (−5): stand on 3, walk 5 to the left, and you land on −2. When the signs are different you're really finding the gap: 5 and 3 differ by 2, and the 5 is negative, so the answer is negative.
Your Turn
Trace the walk with your finger for −4 + 6 and −3 + (−4). Check: 2 and −7. Then use the shortcut: same signs → add the sizes and keep the sign; different signs → subtract the sizes and take the sign of the larger size.
Watch Out
Learn the walk before the shortcut. If you only memorise 'different signs subtract', you'll mix up which sign survives — the picture settles it every time.
Explanation
Subtracting a number is the same as adding its opposite. So 4 − (−3) becomes 4 + 3, which is 7. Think of it as removing a debt — taking away a negative leaves you better off. Whenever two signs sit side by side, tidy them first: minus-minus becomes plus, plus-minus becomes minus.
Your Turn
Tidy the signs first, then compute: 5 − (−2), −1 + (−6), −3 − (−3). Check: 5 + 2 = 7; −1 − 6 = −7; −3 + 3 = 0.
Watch Out
The classic error is 4 − (−3) = 1 — seeing 'minus' and heading left. The remove-a-debt story plus the tidy-the-signs-first routine prevents it.
Explanation
For multiplying and dividing, work out the size first, then the sign. Same signs give a positive answer; different signs give a negative one. Why does negative times negative make positive? If losing $5 a week is −5, then reversing that for 3 weeks — (−3) × (−5) — leaves you $15 better off. Division follows exactly the same sign rules.
Your Turn
Decide the sign first, then the number: (−3) × 7, (−8) ÷ (−2), 5 × (−6), (−40) ÷ 8. Check: −21, 4, −30, −5.
Watch Out
Keep the addition rules and the multiplication rules in separate boxes: −3 + (−3) = −6 (same signs, still negative!) but (−3) × (−3) = 9. Mixing these two rule-sets is the number one error in this chapter.
Explanation
When an expression mixes operations, everyone must agree on the order, or the same question gives different answers. Brackets first, then orders — that's powers and roots — then division and multiplication together from left to right, then addition and subtraction together from left to right. So 5 + 2 × 3 is 5 + 6 = 11 — the multiplication wins over the addition.
Your Turn
Careful with the two 'equal rank' rows — work these strictly left to right: 12 ÷ 4 × 3 and 10 − 3 + 2. Check: (12 ÷ 4) × 3 = 9 (not 12 ÷ 12 = 1), and 9 (not 5).
Watch Out
Reading BODMAS as "D beats M" is a common mistake: division does NOT beat multiplication — they tie and go left to right. Same for A and S.
Explanation
Let's run one from start to finish: 12 − 8 ÷ (−2) × 3 + (−4). Division and multiplication first, left to right: 8 ÷ (−2) = −4, then −4 × 3 = −12. Now tidy the signs: − (−12) becomes + 12, and + (−4) becomes − 4. That leaves 12 + 12 − 4 = 20.
Your Turn
Copy the layout — one operation changed per line, everything else recopied — and evaluate 20 − 12 ÷ (−4) × 5 yourself. Check: 12 ÷ (−4) = −3; −3 × 5 = −15; 20 − (−15) = 35.
Watch Out
Marks are lost to skipped lines, not to wrong rules. Doing two operations in one line is exactly where signs get dropped — slow down and write every line.
Explanation
Before you practise, meet the three traps that catch almost everyone. One: −32 without brackets means minus-the-square — that's −9 — while (−3)2 with brackets is +9. The power only grabs what it touches. Two: order of operations — 5 − 2 × 3 is −1, because the multiplication happens first. Three: don't let the multiplication sign rules leak into addition — adding two negatives stays negative; multiplying two negatives turns positive.
Your Turn
True or false? '−22 = 4', '(−5)2 = 25', '−6 + (−6) = 36'. Check: false (it's −4), true, false (it's −12). Now open the Exercise tab, then do the Quiz.
Watch Out
If you get quiz or exercise questions wrong, come back to this step and work out which of the three traps caught you — it is almost always one of these.
📌 Key Points — this lesson in a nutshell
- Subtracting a negative is adding: a − (−b) = a + b.
- Sign rules for × and ÷: same signs give +, different signs give −.
- BODMAS: Brackets, then powers, then × ÷ left-to-right, then + − left-to-right.
- (−3)2 = 9 but −32 = −9 — the bracket decides what gets squared.
Step 1: Introduction
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