Non-Calculator Strategies (Paper 2 / Paper 5)

On OCR GCSE Mathematics (J560), the middle paper of each tier is the non-calculator paper: Paper 2 (J560/02) on Foundation and Paper 5 (J560/05) on Higher. For 90 minutes and 100 marks you have only your own mental and written methods — no calculator to lean on. This is the paper students most often under-prepare for, precisely because the other four papers let them use a calculator. This lesson builds the by-hand toolkit you need: secure mental and written arithmetic, fraction work, estimation, standard form by hand, and (for Higher) surds. The aim is speed and accuracy without a machine.

This is mostly an AO1 lesson — the non-calculator paper is heavy on accurate standard technique — but estimation and checking are AO3 skills, and explaining or justifying a by-hand result draws on AO2. The encouraging truth is that none of these by-hand skills is beyond any student; they simply need practising regularly so that, on the day, your hand knows the method even under pressure.

Key Reminder: Which Paper Is Non-Calculator?

Tier	Non-calculator paper	Calculator papers
Foundation	Paper 2 (J560/02)	Papers 1 and 3
Higher	Paper 5 (J560/05)	Papers 4 and 6

If you have come from an AQA or Edexcel mock, do not assume "Paper 1 is non-calculator" — on OCR it is the middle paper. Prepare your by-hand skills specifically for Paper 2 (Foundation) or Paper 5 (Higher).

Because the non-calculator paper sits in the middle of your three, it is easy to under-prepare for it: the first paper you sit allows a calculator, which can lull you into calculator habits just before the paper that bans it. Guard against this by deliberately practising whole topics without a calculator in the run-up to the exams, not only the topics that obviously demand it. Fractions, percentages, standard form and (on Higher) surds all appear on the non-calculator paper, and they are exactly the topics where students reach for a calculator out of habit. Train the by-hand versions until they feel as natural as the calculator versions.

Written Methods for the Four Operations

You cannot afford to be slow or shaky on column arithmetic, long multiplication and division — these underpin almost everything else on the paper.

Long multiplication

Work out $34 \times 27$34×27 without a calculator.

Use the column (or grid) method. Splitting $27 = 20 + 7$27=20+7:

$34 \times 7 = 238, \qquad 34 \times 20 = 680.$34×7=238,34×20=680.

Add: $238 + 680 = 918$238+680=918. Answer: $918$918.

A quick sanity check: $34 \times 27 \approx 30 \times 30 = 900$34×27≈30×30=900, and $918$918 is close, so it is reasonable.

Long division

Work out $952 \div 7$952÷7.

Short division: $9 \div 7 = 1$9÷7=1 remainder $2$2; bring down to make $25$25; $25 \div 7 = 3$25÷7=3 remainder $4$4; bring down to make $42$42; $42 \div 7 = 6$42÷7=6. So $952 \div 7 = 136$952÷7=136. Check by reversing: $136 \times 7 = 952$136×7=952. Answer: $136$136.

Decimal arithmetic by hand

For $\times$× and $\div$÷ with decimals, work with the digits and place the point afterwards. For $4.6 \times 0.3$4.6×0.3: ignore the points to get $46 \times 3 = 138$46×3=138; there are two decimal places in the question ($4.6$4.6 has one, $0.3$0.3 has one), so the answer has two: $1.38$1.38. Answer: $1.38$1.38.

For dividing by a decimal, the standard trick is to make the divisor a whole number. To work out $7.2 \div 0.4$7.2÷0.4, multiply both numbers by 10 (which does not change the answer): $72 \div 4 = 18$72÷4=18. Answer: $18$18. Multiplying both numbers by the same power of 10 turns an awkward decimal division into a tidy whole-number one — a reliable non-calculator move.

Mental-arithmetic shortcuts

The non-calculator paper rewards a few quick mental strategies that save time and reduce error:

Shortcut	Example
To $\times 5$×5, multiply by 10 then halve	$48 \times 5 = 480 \div 2 = 240$48×5=480÷2=240
To $\times 9$×9, multiply by 10 then subtract the number	$23 \times 9 = 230 - 23 = 207$23×9=230−23=207
To $\div 5$÷5, double then divide by 10	$90 \div 5 = 180 \div 10 = 18$90÷5=180÷10=18
To $\times 25$×25, multiply by 100 then divide by 4	$16 \times 25 = 1600 \div 4 = 400$16×25=1600÷4=400

These are not tricks for their own sake — they convert a fiddly calculation into one you can do confidently in your head, leaving less room for slips.

Fractions Without a Calculator

Fraction questions are a guaranteed feature of the non-calculator paper. Keep numbers as exact fractions; do not convert to rounded decimals.

Operation	Method
Add / subtract	Common denominator, then add/subtract numerators
Multiply	Multiply numerators, multiply denominators, simplify (cancel first if you can)
Divide	Multiply by the reciprocal of the second fraction ("keep, change, flip")
Of an amount	"of" means multiply

Worked Example: mixed-number division

Work out $2\dfrac{1}{2} \div \dfrac{3}{4}$221​÷43​.

Convert the mixed number: $2\dfrac{1}{2} = \dfrac{5}{2}$221​=25​. Divide by multiplying by the reciprocal:

$\dfrac{5}{2} \div \dfrac{3}{4} = \dfrac{5}{2} \times \dfrac{4}{3} = \dfrac{20}{6} = \dfrac{10}{3} = 3\dfrac{1}{3}.$25​÷43​=25​×34​=620​=310​=331​.

Answer: $3\dfrac{1}{3}$331​. Notice we cancelled and gave the answer as a mixed number in lowest terms — present fractions in the form asked for.

Worked Example: fraction of an amount

Work out $\dfrac{3}{8}$83​ of £64.

Find one eighth, then multiply by 3: $64 \div 8 = 8$64÷8=8, and $8 \times 3 = 24$8×3=24. Answer: £24. (Keep the £ sign in your prose; do not write money inside the maths.)

Percentages Without a Calculator

Percentage questions on the non-calculator paper are made easy by building them from the "friendly" percentages — $10\%$10%, $1\%$1%, $50\%$50% and $25\%$25% — which you can find by simple division.

Percentage	How to find it by hand
$50\%$50%	Halve the amount
$25\%$25%	Halve, then halve again (or divide by 4)
$10\%$10%	Divide by 10
$1\%$1%	Divide by 100
$5\%$5%	Half of $10\%$10%

Worked Example: building a percentage

Work out $35\%$35% of £240 without a calculator.

Build it from $10\%$10% and $5\%$5%: $10\%$10% of 240 is 24, so $30\%$30% is $3 \times 24 = 72$3×24=72; and $5\%$5% is 12 (half of $10\%$10%). Add: $72 + 12 = 84$72+12=84, i.e. £84. Answer: £84. Breaking an awkward percentage into friendly pieces is far safer by hand than trying to multiply by $0.35$0.35 in your head.

Estimation — Rounding to 1 Significant Figure

Estimation questions ("Estimate the value of…") are explicitly non-calculator and reward a clean, fast method: round every number to 1 significant figure, then compute.

Worked Example

Estimate $\dfrac{38.6 \times 5.1}{0.196}$0.19638.6×5.1​.

Round each to 1 s.f.: $38.6 \approx 40$38.6≈40, $5.1 \approx 5$5.1≈5, $0.196 \approx 0.2$0.196≈0.2.

$\dfrac{40 \times 5}{0.2} = \dfrac{200}{0.2} = 1000.$0.240×5​=0.2200​=1000.

Answer (estimate): $1000$1000. Show the rounded values and the working — the marks are for the method, not a precise figure. Estimation is also your built-in check on every non-calculator answer.