IGCSE Maths Year 10: How to Start (Quick Reference)

Most students begin Year 10 IGCSE Maths by copying everything the teacher writes. Two years later they have a beautiful set of notes they never reread. The students who get grade 8 and 9 do something different: from week one they build a one-page quick reference, the sheet they will actually open the night before Paper 2 and Paper 4.

Think of this article as the skeleton of that sheet. For each big idea you need three things: the key result (the formula or fact), when it applies (the trigger in the question), and the classic trap (the mistake that costs the mark). Notes explain; a reference triggers recall under time pressure. You want the second one.

First practical decision: are you sitting Core or Extended? This changes your ceiling (Core caps at grade C / 5, Extended reaches A* / 9) and it changes which results you must know cold. If nobody has told you yet, read Extended vs Core: how to decide before you build anything, because the reference sheet for Extended is roughly twice as long.

Board matters too. On Cambridge 0580 you sit two papers; on Edexcel 4MA1 the structure differs slightly. The maths is 95% the same, but the formula sheet you are given in the exam is not, so check yours early and only memorise what is not printed for you.

These appear in almost every question, so they belong at the top of the sheet.

Indices.

Key results: $a^m \times a^n = a^{m+n}$, $a^m \div a^n = a^{m-n}$, $(a^m)^n = a^{mn}$, and $a^{-n} = \frac{1}{a^n}$, with $a^0 = 1$. When it applies: any time bases are equal. Classic trap: writing $a^m + a^n = a^{m+n}$. You may only add indices when you multiply the powers, never when you add the terms.

Standard form.

Key result: a number as $A \times 10^n$ with $1 \le A < 10$. When it applies: very large or very small numbers, common in the Physics-flavoured questions. Trap: leaving $A$ as $12.4 \times 10^3$ instead of $1.24 \times 10^4$.

Percentages and reverse percentages.

Key result: a $15\%$ increase is a multiplier of $1.15$; the original before a $15\%$ increase is found by dividing by $1.15$, not by subtracting $15\%$. When it applies: the word 'before', 'original', or 'cost price' is the trigger. Trap: the single most common lost mark in the whole syllabus is taking $15\%$ off the new price instead of dividing.

Compound interest.

Key result: $$V = P\left(1 + \frac{r}{100}\right)^n$$ When it applies: growth or depreciation over $n$ periods (use a minus sign inside the bracket for depreciation). Trap: adding simple interest year by year on a calculator and rounding halfway.

Solving linear equations and inequalities.

Trap to write in red: when you multiply or divide an inequality by a negative number, the inequality sign flips. $-2x > 6$ gives $x < -3$, not $x > -3$.

This is the Extended topic that separates a 6 from an 8. Put three tools on your sheet and know exactly when each applies.

Tool 1, factorising.

Use it first, when the numbers are small and friendly. Tool 2, the quadratic formula. Use it when factorising fails or the roots are not whole numbers: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Tool 3, completing the square. Use it when the question says 'find the minimum point' or 'write in the form $(x + p)^2 + q$'.

Worked example.

Solve $2x^2 - 7x + 3 = 0$.

• Try factorising. We need two numbers that multiply to $2 \times 3 = 6$ and add to $-7$: these are $-6$ and $-1$.
• Split the middle term: $2x^2 - 6x - x + 3 = 0$.
• Factor in pairs: $2x(x - 3) - 1(x - 3) = 0$, so $(2x - 1)(x - 3) = 0$.
• Set each factor to zero: $2x - 1 = 0$ gives $x = \frac{1}{2}$, and $x - 3 = 0$ gives $x = 3$.

Check with the formula if you doubt it: $a = 2$, $b = -7$, $c = 3$, so $b^2 - 4ac = 49 - 24 = 25$ and $\sqrt{25} = 5$, giving $x = \frac{7 \pm 5}{4}$, that is $x = 3$ or $x = \frac{1}{2}$. Same answer.

Classic traps to write on the sheet: forgetting the $\pm$ (you lose one whole root and half the marks), sign errors on $b$ when $b$ is negative (that $-(-7)$ becomes $+7$), and stopping at $(2x-1)(x-3)=0$ without stating the two values of $x$. For a deeper drill on this, see how to solve IGCSE quadratic equations.

The single most useful decision on the sheet is a two-line flowchart for triangles.

Right-angled triangle?

Use Pythagoras and SOHCAHTOA. Key results: $a^2 + b^2 = c^2$ for the sides, and $\sin\theta = \frac{\text{opp}}{\text{hyp}}$, $\cos\theta = \frac{\text{adj}}{\text{hyp}}$, $\tan\theta = \frac{\text{opp}}{\text{adj}}$ for the angles. When it applies: you see a right angle marked, or you can drop one in.

Not right-angled?

Use the sine rule or the cosine rule. Sine rule: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ Use it when you have a matching side-and-angle pair. Cosine rule: $$a^2 = b^2 + c^2 - 2bc\cos A$$ Use it when you have two sides and the angle between them, or all three sides. Area of any triangle: $\text{Area} = \frac{1}{2}ab\sin C$, used when you know two sides and the included angle.

Classic traps for the sheet:

• Calculator in degrees. If your trig answers are absurd, your calculator is in radians. Check the D on the display before Paper 2.
• Rounding early. Keep full accuracy in the calculator and round only the final answer, usually to 3 significant figures unless told otherwise.
• Bearings measured wrong. Bearings are always measured clockwise from north and written with three figures, so due east is $090$, not $90$.

Circle theorems are pure recall: the angle in a semicircle is $90$, the angle at the centre is twice the angle at the circumference, and opposite angles in a cyclic quadrilateral sum to $180$. When a question says 'give reasons', the reason is worth a mark on its own, so name the theorem.

Knowing the maths is half the job. The other half is reading the paper like the examiner wrote it, which they did.

Command words are instructions, not decoration.

'Work out' and 'calculate' expect a number with working. 'Show that' means the answer is given, so you must produce every line that reaches it; a bare final number scores zero. 'Write down' expects no working, so do not waste time. 'Give an exact answer' or 'in terms of $\pi$' forbids the decimal button.

Method marks survive wrong answers.

IGCSE mark schemes split marks into M (method) and A (accuracy). If you write the correct method and slip on the arithmetic, you still bank the M marks, but only if the working is on the page. This is why 'no working, wrong answer' is the worst possible outcome. Read how mark schemes work so you know exactly where the marks live.

Manage the two papers differently.

The non-calculator paper rewards clean fractions and surds; the calculator paper rewards typing carefully and using the memory button instead of copying decimals. On both, the marks per question are printed in the margin: a 4-mark question wants roughly four minutes and four visible steps.

Use past papers as diagnosis, not decoration.

From the first month of Year 10, do one topic's worth of real past-paper questions and mark them against the official scheme. Every mark you drop is one line for your reference sheet. That feedback loop, not rereading notes, is what moves a grade. How to use IGCSE past papers walks through the method.

The reference sheet only works if you revisit it, so build revisiting into the calendar from the start.

• Autumn of Year 10. Lock in number, algebra basics, and ratio. Add each key result to the sheet the week you meet it, with its trap.
• Spring of Year 10. Quadratics, functions, and the trig flowchart. This is the hardest new material; do one past-paper topic set every fortnight.
• Summer of Year 10. Geometry, mensuration, and probability. Sit your first full past paper under time to see how the reference holds up under pressure.
• Year 11. No new sheet. You now condense the two-year sheet down to the twenty results you still get wrong, and drill only those.

The habit that ties it together is a weekly ten-minute review: read the sheet, cover it, and rewrite three results from memory. If you can reproduce a result and its trap without looking, it leaves the active sheet. What remains, shrinking each week, is exactly what you open the night before the exam.