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Paper 1 is where grade boundaries are won or lost. Without a calculator, every mark depends on clean working and confident mental arithmetic. These techniques are drilled in our tuition sessions and will help you pick up marks other students leave behind.
In the non-calculator paper, hesitation costs marks. You need to know multiplication tables to 12 instantly, but you also need reliable backup methods for larger numbers. Partitioning is the safest: split 17 × 8 into (10 × 8) + (7 × 8) = 80 + 56 = 136. For division, use equivalent fractions: 84 ÷ 6 is the same as (60 ÷ 6) + (24 ÷ 6) = 10 + 4 = 14.
Practise these daily for five minutes. Speed builds only through repetition, not through reading about methods. If you are unsure of a table, write it out before the exam and read it in the waiting room. It is not cheating; it is preparation.
Exam tip: If a question asks for 15% of £240, find 10% (£24) and 5% (£12), then add. Never attempt 0.15 × 240 in your head unless you are completely confident. Break it down every time.
Fractions appear on every non-calculator paper. The key is to find a common denominator before you add or subtract, and to cancel factors before you multiply. Never jump to decimal conversion unless the question asks for it — decimals introduce rounding errors and lose method marks.
Step 1 — Find a common denominator.
The lowest common multiple of 8 and 12 is 24. (M1)
Step 2 — Rewrite each fraction.
3/8 = 9/24 and 5/12 = 10/24.
Step 3 — Add and simplify.
9/24 + 10/24 = 19/24. Since 19 is prime, the fraction is already in its simplest form. (A1)
A common mistake is to add the numerators and denominators separately to get 8/20. The examiner awards no marks for this because it shows no understanding of equivalent fractions.
For multiplying fractions, always cancel across first. 4/15 × 5/8 becomes 1/3 × 1/2 = 1/6 after cancelling the 4 with the 8 and the 5 with the 15. This keeps numbers small and reduces arithmetic errors. For more algebra practice, see our free revision packs.
Examiners expect you to check whether your answer is sensible. If you calculate that a car travels 3,000 kilometres in two hours, something has gone wrong. Estimation is the fastest check: round every number to one significant figure, do the rough calculation, and compare.
Step 1 — Round to one significant figure.
47.8 → 50, 6.12 → 6, 0.198 → 0.2. (M1)
Step 2 — Compute the estimate.
(50 × 6) / 0.2 = 300 / 0.2 = 1,500. (A1)
The exact answer is approximately 1,476, so the estimate is close. If your exact answer is 147.6, the estimation immediately flags a decimal-point error.
Bounds questions also appear on the non-calculator paper. Remember that if a length is given as 6 cm to the nearest centimetre, the lower bound is 5.5 cm and the upper bound is 6.5 cm. When calculating a maximum area, use the upper bound for every measurement involved.
Algebra without a calculator is mostly about pattern recognition. Factorising quadratics in the form x² + bx + c requires two numbers that multiply to c and add to b. If c is positive, the numbers have the same sign. If c is negative, they have opposite signs.
Step 1 — Identify the factor pair.
We need two numbers that multiply to −24 and add to −5. The pair is −8 and +3. (M1)
Step 2 — Write the factorised form.
x² − 5x − 24 = (x − 8)(x + 3). (A1)
Always expand your answer mentally to check: (x − 8)(x + 3) = x² + 3x − 8x − 24 = x² − 5x − 24. This takes five seconds and catches sign errors.
For expanding double brackets, use the FOIL method systematically: First, Outside, Inside, Last. Write out all four terms before combining them. Skipping the intermediate step is the most common way students lose method marks in non-calculator algebra.
Indices rules are tested heavily on Paper 1 because they require no calculator. The three rules to memorise are: am × an = am+n; am ÷ an = am−n; and (am)n = amn. A negative index means reciprocal: a−n = 1 / an.
Standard form questions usually ask you to multiply or divide two numbers in the form a × 10n. Multiply the a values, multiply the powers of ten using the index rule, then convert back to standard form if the result is outside the range 1 to 10. For example, (3 × 104) × (2 × 105) = 6 × 109. Straightforward, but only if the rules are automatic.
Subtracting a negative becomes adding, and multiplying two negatives gives a positive. Write every step out. Half of all lost algebra marks come from rushing the sign handling.
0.3 × 0.2 is not 0.6. Count decimal places: one plus one equals two, so the answer is 0.06. If in doubt, rewrite as fractions: 3/10 × 2/10 = 6/100.
Brackets, Indices, Division and Multiplication, Addition and Subtraction. A question like 3 + 4 × 2 gives 11, not 14. Underline the operation you will do first before you write anything.
Examiners cannot award method marks for work done only on a calculator. Show every line of working, even if it feels obvious. A correct answer with no working often scores zero on show-that questions.
Non-calculator skills improve fastest with structured repetition. Our STEM Mission Boxes include classroom activities that embed mental arithmetic and estimation into real-world scenarios. Schools receive ready-to-use worksheets, teacher guidance and extension problems that build the same number fluency tested in Paper 1.
Download our free AQA, Edexcel and OCR GCSE maths starter packs — each with thirty worked questions and board-specific topic rankings. If you want one-to-one support, book a free discovery call with a DBS-checked tutor.