Integers and Order of Operations

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.

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.

Raising a negative number to a power is repeated multiplication, so the sign-counting rule applies directly. $(-3)^4$ means $-3 \times -3 \times -3 \times -3$ — four negatives, an even count, so the answer is positive $81$. But $(-3)^3$ has three negatives, an odd count, so it is negative $-27$.

The trap to meet head-on is the difference between $(-4)^2$ and $-4^2$. They look almost identical but mean different things:

• $(-4)^2$ means the whole of $-4$ is squared: $-4 \times -4 = 16$.
• $-4^2$ means only the 4 is squared, and the minus is applied afterwards: $-(4 \times 4) = -16$.

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.

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.

So $50 - 3 \times (12 - (5 + 2))^2$ goes: inner bracket $5 + 2 = 7$; outer bracket $12 - 7 = 5$; index $5^2 = 25$; multiplication $3 \times 25 = 75$; subtraction $50 - 75 = -25$.

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 $\frac{8 + 4}{3}$ you finish the top before dividing.

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 $(-3)^3 + 24 \div (-4) - (-7)$.

• Indices first: $(-3)^3 = -27$ (three negatives, odd, so negative).
• Division next: $24 \div (-4) = -6$ (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 $-33 + 7 = -26$.

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.

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 $-6\degree\text{C}$ rises by $11\degree$ — that is $-6 + 11 = 5\degree\text{C}$. The difference between $-6\degree\text{C}$ and $5\degree\text{C}$ is $5 - (-6) = 11\degree$, a subtraction of one reading from the other.

The same care helps with bank balances, heights above and below sea level and game scores. A balance of -\40$that receives a$$25$payment becomes$-40 + 25 = -$15$— still overdrawn. A diver at$-18$m who rises$7$m is at$-18 + 7 = -11$ m. Read slowly, decide what the sign means, then choose the operation with confidence.

Solid integer and order-of-operations skills make a huge amount of later maths easier:

• Algebra substitutes negative values into expressions and formulae constantly — $2x^2 - 5x$ 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.
• Standard form relies on negative indices, which are simply negative integers used as powers.
• 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.