Paper 1 Masterclass
PAPER 1 MASTERCLASS • EXAM TECHNIQUE
Non-calculator techniques, mental methods, extended response strategy & time management — everything you need to squeeze every mark out of Paper 1, Foundation or Higher.
Paper 1 at a glance
Paper 1 is the non-calculator paper. You sit it alongside Papers 2 and 3, but everything on Paper 1 has to be done by hand.
FeatureDetail
Time1 hour 30 minutes
Marks80
Weighting33⅓% of the qualification
CalculatorNot permitted
ContentAny topic on the specification
TiersFoundation grades 1–5; Higher grades 4–9
KEY POINT
Paper 1 rewards fluency with by-hand methods and exact-form working. A strong Paper 1 performance comes from planning, method marks, and keeping things exact — not from brute arithmetic speed.
The non-calculator mindset
Every Paper 1 question has a method that works by hand. If what you're writing feels like it needs a calculator, stop — you're using the wrong method.
THREE RULES
1. Keep answers exact — fractions, surds and multiples of π, not decimals.
2. Show every step of working — method marks are awarded per line.
3.Simplify as you go — don't let numbers grow large when you can cancel early.
EXAMINER ALERT
Candidates often attempt a method that requires a calculator — for example evaluating a complicated decimal, or a trig calculation at a non-standard angle — and get stuck. If a method feels calculator-intensive on Paper 1, there is an easier non-calculator route — find it.
Core skill: Fraction arithmetic
Fractions appear on every Paper 1. You must handle all four operations cleanly and without a calculator.
OperationMethodExample
Add / SubtractCommon denominator, then combine numerators2/3 + 1/4 = 8/12 + 3/12 = 11/12
MultiplyCross-cancel first, then multiply numerators and denominators3/8 × 4/9 = (1/2)×(1/3) = 1/6
DivideMultiply by the reciprocal ("keep, change, flip")2/5 ÷ 4/15 = 2/5 × 15/4 = 3/2
Mixed numbersConvert to improper first, then operate2⅓ = 7/3
CROSS-CANCEL FIRST
Always cancel common factors before multiplying. Multiplying first gives you large numbers that must then be cancelled anyway — and introduces arithmetic error.
Core skill: Long multiplication & division
GRID METHOD (×)
Split each number by place value, multiply each part, then sum. Reliable for 2-digit by 2-digit. 37 × 24
= (30+7)(20+4)
= 600 + 120 + 140 + 28
= 888
BUS-STOP METHOD (÷)
Work left to right, carrying remainders. Add a decimal point and zeros for decimal answers. 672 ÷ 4
= 168
(4 into 6 = 1 r2, 4 into 27 = 6 r3, 4 into 32 = 8)
EXAM TIP
For longer multiplications (3-digit by 2-digit), the column method is faster and uses less page. Pick whichever you're more accurate with — speed matters less than correctness on Paper 1.
Core skill: Percentages — mental decomposition
Every percentage can be built from 10%, 5%, 1% — and all three are quick mental divisions.
Building blockHow to find itOf £240
10%Divide by 10 (shift decimal)£24
5%Half of 10%£12
1%Divide by 100£2.40
50%Half£120
25%Quarter£60
WORKED DECOMPOSITION
Find 17% of £240
= 10% + 5% + 2% of £240
= £24 + £12 + £4.80
= £40.80
REVERSE PERCENTAGES
Reverse questions divide by the multiplier, not the percentage. After a 20% increase, the multiplier is 1.2 — so reverse by dividing by 1.2 (which on Paper 1 means multiplying by 10/12 = 5/6).
Core skill: Standard form by hand
On Paper 1 you must add, subtract, multiply and divide numbers in standard form without a calculator.
OperationMethod
MultiplyMultiply coefficients; add the powers of 10
DivideDivide coefficients; subtract the powers of 10
Add / SubtractMatch the powers of 10 first , then add or subtract the coefficients
WORKED EXAMPLE
(3 × 10⁵) + (4 × 10⁴)
= (3 × 10⁵) + (0.4 × 10⁵) — matched powers
= 3.4 × 10⁵
FINAL-FORM CHECK
A standard-form answer must have the coefficient A in the range 1 ≤ A < 10. Always adjust the power of 10 at the end if needed. 24 × 10⁶ = 2.4 × 10⁷.
Core skill: Surds HIGHER ONLY
Surds are a Paper 1 staple at Higher. The technique is always the same: factor out the largest square number.
SURD LAWS
√a × √b = √(ab)
√a ÷ √b = √(a/b)
√(a²b) = a√b
SIMPLIFYING
√50 = √(25 × 2) = 5√2
√72 = √(36 × 2) = 6√2
√200 = √(100 × 2) = 10√2
RATIONALISING
6/√3
= 6/√3 × √3/√3
= 6√3/3
= 2√3
EXAM TIP
When rationalising a/(b + √c), multiply by the conjugate (b − √c). The denominator becomes b² − c — a rational number.
Core skill: Recurring decimals HIGHER ONLY
Converting a recurring decimal into a fraction is a Paper 1 classic. The method is always the same: multiply by a power of 10 that matches the recurring block, then subtract.
METHOD
1. Let x = the recurring decimal.
2. Multiply by 10ⁿ, where n is the length of the repeating block.
3. Subtract the original equation — the recurring tail cancels.
4. Solve for x as a fraction; simplify.
Express 0.2̇7̇ as a fraction in its simplest form.
- MRecurring block is "27" — length 2, so multiply by 10² = 100.
- CLet x = 0.272727... → 100x = 27.272727...
- CSubtract: 100x − x = 27.272727... − 0.272727...
- C99x = 27 → x = 27/99
- ESimplify by 9: x = 3/11
EXAM TIP
The denominator comes out as a string of 9s (one per recurring digit): 9, 99, 999, ... If only part of the decimal recurs — for example 0.16̇ = 0.16666... where only the 6 repeats — multiply by two different powers of 10 (here 10x and 100x) so the tails still cancel when you subtract.
Core skill: Exact trig values
Exact-values trig questions appear on Paper 1 at both tiers (spec G21). The values come from two special triangles you must know cold.
θ0°30°45°60°90°
sin θ01/2√2/2√3/21
cos θ1√3/2√2/21/20
tan θ0√3/31√3—
Values to memorise
These come up so often on Paper 1 that calculating them each time costs you marks. Learn them.
SQUARES & CUBES
Squares to 15²:
1, 4, 9, 16, 25, 36, 49, 64,
81, 100, 121, 144, 169, 196, 225
Cubes to 10³:
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
PRIMES UNDER 50
2, 3, 5, 7, 11, 13, 17, 19,
23, 29, 31, 37, 41, 43, 47
FractionDecimalPercentage
1/20.550%
1/40.2525%
1/50.220%
1/80.12512.5%
1/30.333...33⅓%
The exact-form rule
Most Paper 1 algebra, geometry and trig answers are expected in exact form. This means do not convert to a decimal.
QuantityExact formWhat loses marks
πLeave as π (e.g. 18π)56.55...
SurdsLeave as √2, 5√31.414..., 8.660...
FractionsKeep as a fraction0.333... or 0.33
Trig values1/2, √3/2, √2/20.5, 0.866, 0.707
EXAMINER ALERT
A common Paper 1 error is converting π to 3.14 at the start and dragging an approximation through the whole calculation. Every step loses a little precision and the final answer falls outside the accept range. Mark schemes on exact-form questions will often say "do not allow use of a numerical value for π".
Worked example: Exact sector area
01Paper 1 • Exact form
A sector has radius 12 cm and angle 45° at the centre. Find the exact area of the sector. [3 marks]
M
Area of sector = (θ / 360) × π × r²
C
= (45 / 360) × π × 12²
C
= (1/8) × π × 144
C
= 144π / 8 = 18π
E
Area = 18π cm² (exact form — do not write 56.5)
The 5-mark question protocol
Extended-response questions (5 or more marks) reward planning, not speed. Use this 5-step approach on every one.
1Read the whole question before writing.
2Identify the asked-for quantity (units? exact form? which letter?).
3Plan the method — write a short "M:" line if helpful.
4Show every substitution, every rearrangement, every evaluation.
5Label the final answer clearly and in the requested form.
WHY THIS WORKS
On a 5-mark question, the final answer is typically worth 1 mark. The other 4 are method marks — you can score them even if you make an arithmetic slip at the end, provided the working is laid out.
Worked example: Multi-step fractions
Paper 1 • Non-calculator fractions02
Without using a calculator, work out (3/4 + 1/6) ÷ 2⅓. Give your answer as a fraction in its simplest form. [4 marks]
M
Inside bracket: common denominator 12. Convert 2⅓ to improper fraction. Divide = multiply by reciprocal.
C
3/4 + 1/6 = 9/12 + 2/12 = 11/12
C
2⅓ = 7/3
C
(11/12) ÷ (7/3) = (11/12) × (3/7)
C
Cross-cancel 3 and 12: = 11/4 × 1/7 = 11/28
E
Answer = 11/28 (already in simplest form — 11 is prime and does not divide 28)
Time management
You have 90 minutes for 80 marks — that is just over 1 minute per mark.
TARGET BUDGET
1-mark question: ~60 seconds
3-mark question: ~3–4 minutes
5-mark question: ~5–6 minutes
Final check: 5 minutes
SKIP AND RETURN
If you haven't started after 30 seconds, move on.
Mark skipped questions with a dot and come back.
Do the easy parts of every question — 1-mark definitions don't care what came before.
EXAMINER ALERT
Examiners consistently see students lose 10+ easy marks at the end of Paper 1 because they spent too long on a single difficult question earlier. Every mark counts the same — don't sacrifice five 1-markers for one hard 5-marker.
Practice question HIGHER
A right-angled triangle has two shorter sides (legs) of length 5 cm each.
Find the exact length of the hypotenuse, giving your answer in the form a√b where a and b are integers and b is as small as possible. [3 marks]
M
Pythagoras: hypotenuse² = 5² + 5²
C
h² = 25 + 25 = 50
C
h = √50 = √(25 × 2) = 5√2
E
Hypotenuse = 5√2 cm — do not write 7.07
Paper 1 — top examiner alerts
ALERT 1 — DECIMAL APPROXIMATION
Converting π to 3.14 or √2 to 1.41 on any exact-form question will lose the final-answer mark. Mark schemes explicitly reject numerical approximations where exact form is requested.
ALERT 2 — ROUNDING MID-CALCULATION
Rounding partway through a multi-step calculation compounds error. Keep exact values (fractions or surds) all the way through, and round only on the final line.
ALERT 3 — PANIC-SKIPPING WHOLE QUESTIONS
Even "hard" questions have accessible parts — a 1-mark definition, a "write down" instruction, or part (a) of a 6-mark question. Attempt every part on the paper. Blank answers score zero; attempts can score method marks.
Paper 1 — summary
Key Points
Paper 1 is 90 minutes, 80 marks, non-calculator — any topic on the spec.
Keep answers in exact form (π, surds, fractions) — never convert to decimal unless asked.
Build percentages from 10%, 5%, 1% ; add and subtract standard form by matching powers .
Use the 5-mark protocol : read, identify, plan, show every step, label the answer.
Budget ~1 minute per mark . Skip-and-return beats over-committing.
MEMORISE
Squares to 15², cubes to 10³, primes under 50, fraction↔decimal↔% equivalents, and the exact trig values table.
FLUENT BY HAND
Fractions (four operations), long multiplication (grid), long division (bus-stop), standard form arithmetic, surd simplification.
THE ONE TIP
Every method mark you show is a mark you can't lose to an arithmetic slip. Write the method every time, even when the answer seems obvious — on a Paper 1, working is the insurance policy against a bad day with the numbers.
Next: Paper 1 at a glance