Detailed notes on Number for Cambridge Lower Secondary Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.
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Integers and Order of Operations — frequently asked questions
The things students keep getting wrong in this sub-topic, answered.
Integers and Order of Operations — Cambridge Lower Secondary Maths, Grade 8
You can already calculate with negatives and follow BIDMAS. Stage 9 asks for genuine fluency — handling layered brackets, indices on negative bases, and signed numbers woven through multi-step problems without ever losing track of a sign.
At a glance
An integer is a whole number — positive, negative or zero, never a fraction.
Two signs sitting together combine, so a−(−b)=a+b every time.
Same signs multiply or divide to a positive; different signs give a negative.
Count the negatives in a product: even count → positive, odd count → negative.
(−4)2=16 but −42=−16 — brackets decide what the index acts on.
BIDMAS sets the order: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
Division and multiplication rank equally — always work strictly left to right.
A quick sense-check on the sign and size of an answer catches most slips.
What you’ll learn
Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.
Stage 9 — Calculate fluently with positive and negative integers across all four operations and longer chains.
Stage 9 — Apply the order of operations to multi-step calculations with several layers of brackets and indices.
Stage 9 — Use negative numbers confidently inside word problems and reasoning tasks.
Stage 9 — Estimate and sense-check signed calculations for a sensible sign and size.
Fluent arithmetic with integers
At Stage 9 the four sign rules should run automatically — even inside long calculations.
By Stage 9 the number line is no longer something you need to picture step by step — the sign rules should run automatically. The aim now is fluency under pressure: keeping every sign correct even in a long, busy calculation.
Two ideas still carry all of it. For adding and subtracting, two signs sitting next to each other combine into one — a plus and a minus make a minus, two minuses make a plus. For multiplying and dividing, look only at the two signs: same signs give a positive, different signs give a negative.
Four small rules cover every signed calculation you will meet.
The skill that grows at Stage 9 is stamina — staying accurate across a calculation with five or six operations. Take it one step at a time and rewrite the whole line each time; that single habit removes nearly every careless slip.
Two signs together combine: .
Indices on negative bases
A negative base to an even power is positive; to an odd power it stays negative.
Raising a negative number to a power is repeated multiplication, so the sign-counting rule applies directly. (−3)4 means −3×−3×−3 — four negatives, an even count, so the answer is . But has three negatives, an odd count, so it is .
Order of operations with layered brackets
BIDMAS still rules — the Stage 9 step up is several layers of brackets and indices at once.
The order of operations has not changed: BIDMAS — Brackets, Indices, Division and Multiplication, then Addition and Subtraction. What grows at Stage 9 is the number of layers in a single calculation.
You will meet nested brackets and indices acting on whole bracketed expressions. Always work from the innermost bracket outwards, settling each layer before you move on.
Peel the brackets from the inside, then indices, then the rest.
So 50− goes: inner bracket ; outer bracket ; index ; multiplication ; subtraction .
Combining signs, powers and order
Real Stage 9 calculations weave all three ideas together — one step per line.
The toughest Stage 9 calculations weave together negative numbers, indices and a multi-step order. The cure is the same every time: one step per line, rewriting the whole calculation as you go.
Try (−3)3+24÷(−4)−(−.
Negative numbers inside problems
Word problems hide signed arithmetic — read carefully and decide what each sign means.
Stage 9 questions often dress signed arithmetic up as a real situation. The skill is deciding what a negative number means in context, and which operation the question is really asking for.
Temperatures are a classic. A reading of −6°C rises by 11° — that is −6+11=. The between and is , a subtraction of one reading from the other.
Where you'll use this next
Fluent integer work underpins almost every topic in Stage 9 and beyond.
Solid integer and order-of-operations skills make a huge amount of later maths easier:
Algebra substitutes negative values into expressions and formulae constantly — 2x2−5x when x=−3 is a sign-and-power question in disguise.
Coordinate geometry uses negative coordinates on both axes, and gradients are differences that can be negative.
relies on negative indices, which are simply negative integers used as powers.
Quick recap
An integer is a whole number — positive, negative or zero.
Two signs together combine, so subtracting a negative is the same as adding.
Same signs multiply or divide to a positive; different signs to a negative.
An even power of a negative base is positive; an odd power is negative.
(−4)2=16 but −4 — brackets decide what is squared.
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Where you'll use this next
a−(−b)=a+b
Same signs multiply or divide to a positive answer.
Different signs multiply or divide to a negative answer.
Rewrite the full line after each step to stay accurate.
×
−3
positive
81
(−3)3
negative
−27
Even powers turn a negative base positive; odd powers keep it negative.
The trap to meet head-on is the difference between (−4)2 and −42. They look almost identical but mean different things:
(−4)2 means the whole of −4 is squared: −4×−4=16.
−42 means only the 4 is squared, and the minus is applied afterwards: −(4×4)=−16.
The brackets decide what the index acts on. Whenever you raise a negative number to a power, wrap it in brackets so there is no doubt at all.
A power is repeated multiplication, so the sign rule applies.
Even power of a negative base → positive answer.
Odd power of a negative base → negative answer.
(−4)2=16 but −42=−16 — brackets change the meaning.
3
×
(12−
(5+
2))2
5+2=7
12−7=5
52=25
3×25=75
50−75=−25
Remember that division and multiplication rank equally — you work them left to right, never all division first. The same holds for addition and subtraction. A fraction bar acts like a hidden bracket: in 38+4 you finish the top before dividing.
Division and multiplication rank equally — go left to right.
7
)
Indices first:(−3)3=−27 (three negatives, odd, so negative).
Division next:24÷(−4)=−6 (different signs).
The line is now −27+(−6)−(−7).
Tidy the signs:−27−6+7.
Add and subtract left to right:−27−6=−33, then −33+7=−26.
Each line does one job — the answer is hard to lose this way.
When you finish, glance back: does the sign feel right, and is the size sensible? If you expected a small negative and got a large positive, a sign has slipped somewhere.
Mix of signs, powers and order — do one operation per line.
Indices and brackets are settled before ×, ÷, + and −.
Tidy double signs as soon as the line allows it.
Sense-check the final sign and size of the answer.
5°
C
difference
−6°C
5°C
5−(−6)=11°
Subtracting one reading from the other gives the gap between them.
The same care helps with bank balances, heights above and below sea level and game scores. A balance of -\40thatreceivesa$25paymentbecomes-40 + 25 = -$15—stilloverdrawn.Adiverat-18mwhorises7misat-18 + 7 = -11$ m. Read slowly, decide what the sign means, then choose the operation with confidence.
Decide what a negative number means in the situation.
A difference is one value subtracted from another.
Temperatures, balances, depths and scores all use signed arithmetic.
Read slowly, then choose the operation deliberately.
Standard form
In everyday life, signed calculations describe temperatures, bank balances, elevation and any multi-step calculator problem.
If a later topic feels shaky, it is often a sign rule or a BIDMAS step underneath that needs a quick refresh. Come back to this guide whenever you need it — accuracy beats speed every time.
Algebra substitutes negative numbers into expressions and powers.
Coordinate geometry uses negative values and negative gradients.
Standard form uses negative integers as indices.
Strong integer fluency smooths every later topic.
2
=
−16
BIDMAS sets the order; work nested brackets from the inside out.
Sense-check the sign and size of every answer.
8
−
5=
3,3+
3=
6,6−
6=
0
)2
Step-by-step solution
Step 1
Brackets first.
7−4=3
Step 2
Indices next: square the result.
32=9
Step 3
Multiplication before addition.
2×9=18
Step 4
Finally the addition.
4+18=22
Answer
22
−
24
Step-by-step solution
Step 1
(−2)4 squares the whole of −2 four times. Four negatives is an even count, so the answer is positive.
(−2)4=−2×−2×−2×−2=16
Step 2
−24 raises only the 2 to the power, then applies the minus afterwards.
−24=−(2×
Step 3
The brackets decide what the index acts on, so the two results have opposite signs.
Answer
(−2)4=16 and −24=−16
Step 1
'How much warmer' is a difference — subtract the dawn reading from the midday reading.
6−(−7)
Step 2
Two minus signs together make a plus.
6−(−7)=6+7
Step 3
Now add.
6+7=13
Answer
13°C warmer
+
40÷
(8−
(5+
1))−
(−4)
Step-by-step solution
Step 1
Inner bracket first.
5+1=6
Step 2
Outer bracket next.
8−6=2
Step 3
Indices: (−3)3 has three negatives, an odd count, so it is negative.
(−3)3=−27
Step 4
Division: the result is now divided by 2.
40÷2=20
Step 5
Tidy the double sign, then add and subtract left to right.
−27+20+4=−3
Answer
−3
3
+
1
13
▼
Why it happens
Without brackets, the minus looks like it is part of the number being squared.
How to avoid it
−42 squares only the 4: −(4×4)=−16. Only (−4)2 squares the whole −4 to give 16.
6×3=18
Count every negative sign: three negatives is odd, so the answer is negative — here −6.
2×
2×
2)=
−16
Integers and Order of Operations — Cambridge Lower Secondary Mathematics — Stage 9 Revision Notes & Practice | Tutopiya