Integers and Order of Operations

By now the number line is a familiar friend. At Stage 8 the aim is fluency: calculating with integers quickly and without second-guessing the signs.

Two ideas carry all of it. First, 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. So $8 + (-5) = 3$ and $8 - (-5) = 13$.

Second, multiplying and dividing — look only at the two signs. Same signs give a positive, different signs give a negative.

The new skill is keeping all of this steady inside a longer calculation. Take it one operation at a time and rewrite the whole line after each step — that habit alone removes most slips.

Stage 8 questions often string several signed numbers together. The reliable approach is to tidy first, calculate second.

Look at $6 + (-9) - (-4) + (-2)$. Before doing any arithmetic, replace each pair of signs:

• $+ (-9)$ becomes $- 9$.
• $- (-4)$ becomes $+ 4$.
• $+ (-2)$ becomes $- 2$.

The line is now the much friendlier $6 - 9 + 4 - 2$. Working left to right: $6 - 9 = -3$, then $-3 + 4 = 1$, then $1 - 2 = -1$.

The same care pays off with multiplication chains. For $-2 \times -3 \times -1 \times 2$, ignore the digits for a moment and count the negative signs: there are three, an odd number, so the answer is negative. The digits give $2 \times 3 \times 1 \times 2 = 12$, so the result is $-12$.

Raising a negative number to a power is just repeated multiplication, so the sign-counting rule applies directly.

$(-2)^4$ means $-2 \times -2 \times -2 \times -2$ — four negative signs, an even count, so the answer is positive $16$. But $(-2)^3$ has three negatives, an odd count, so it is negative $-8$.

There is one trap worth meeting head-on. $(-3)^2$ and $-3^2$ look almost the same but mean different things:

• $(-3)^2$ means the whole of $-3$ is squared: $-3 \times -3 = 9$.
• $-3^2$ means only the 3 is squared, and the minus is applied afterwards: $-(3 \times 3) = -9$.

The brackets decide what the index acts on. Whenever you square a negative number, wrap it in brackets so there is no doubt.

The order of operations has not changed: BIDMAS — Brackets, Indices, Division and Multiplication, then Addition and Subtraction. What changes at Stage 8 is the difficulty of the calculations.

You will meet nested brackets — brackets inside brackets — and powers that act on whole bracketed expressions. Always work from the innermost bracket outwards.

So $4 + 2 \times (9 - (3 + 1))^2$ goes: inner bracket $3 + 1 = 4$; outer bracket $9 - 4 = 5$; index $5^2 = 25$; multiplication $2 \times 25 = 50$; addition $4 + 50 = 54$.

Remember that division and multiplication rank equally — you work them left to right, not all division first. The same holds for addition and subtraction.

The hardest Stage 8 calculations weave together negative numbers, indices and a multi-step order. The cure is the same every time: one step per line.

Try $(-2)^3 + 12 \div (-4) - (-5)$.

• Indices first: $(-2)^3 = -8$ (three negatives, odd, so negative).
• Division next: $12 \div (-4) = -3$ (different signs).
• The line is now $-8 + (-3) - (-5)$.
• Tidy the signs: $-8 - 3 + 5$.
• Add and subtract left to right: $-8 - 3 = -11$, then $-11 + 5 = -6$.

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.

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

• Algebra substitutes negative values into expressions and formulae constantly — $5x^2$ 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.
• Sequences include terms that decrease into the negatives, with rules built on signed arithmetic.
• 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. Come back to this guide whenever you need it — accuracy beats speed every time.