Every reference card

1. Number facts to know on sight

Prime numbers under 100

There are 25 of them:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

The rules that catch children out:

• 1 is NOT prime (a prime has exactly two factors; 1 has one).
• 2 is prime — and it's the only even prime.
• A prime has exactly two factors: 1 and itself.

The fake primes. These look prime and are the examiner's favourite trap:

Square numbers (know to 15²)

Cube numbers (know to 10³)

Fibonacci sequence

Each term is the sum of the previous two:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

Start 1, 1. Then 1+1=2, 1+2=3, 2+3=5, 3+5=8. If a sequence's third term equals its first two added, test Fibonacci.

Triangular numbers

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, …

Differences go up by one each time (+2, +3, +4, +5…). They're the dot-triangle numbers: 1 dot, then 3, then 6.

Divisibility tests

2. Roman numerals

The seven letters: I=1, V=5, X=10, L=50, C=100, D=500, M=1000. Memory hook: I Value Xylophones Like Cows Do Milk.

The subtraction rule. A smaller letter before a larger one means subtract:

• IV = 4 (not IIII), IX = 9, XL = 40, XC = 90, CD = 400, CM = 900

Reading a long one — work left to right in chunks: MMXXVI → MM (2000) + XX (20) + V (5) + I (1) = 2026

Rules examiners test:

• Never four of the same letter in a row (so 40 is XL, never XXXX)
• Only I, X, C are ever used for subtraction (never V, L, D)
• I only subtracts from V and X; X only from L and C; C only from D and M

3. Fractions, decimals and percentages — the equivalence table

The single most useful table on this page. Learn every row both directions.

Converting, in one line each:

• Fraction → decimal: divide top by bottom (3 ÷ 8 = 0.375)
• Decimal → percentage: × 100 (0.375 → 37.5%)
• Percentage → fraction: over 100, then simplify (45% = 45/100 = 9/20)
• Decimal → fraction: say it aloud — "nought point four five" = 45 hundredths = 45/100 = 9/20

Percentages by building from 10% (faster than long multiplication in an exam):

35% of 240 → 10% = 24 → 30% = 72 → 5% = 12 → 84

4. Units of measurement

The direction rule that prevents most errors: converting to a smaller unit means you need more of them → multiply. To a bigger unit → divide.

4,500 ml → litres. Litres are bigger → divide → 4.5 L ✓

The area trap. 1 m² is 10,000 cm², not 100. Because a square metre is 100 cm × 100 cm.

5. Time — 12-hour and 24-hour

The two rules:

1. Afternoon/evening: add 12. 3:15 pm → 3 + 12 = 15:15
2. Midnight is 00:00, noon is 12:00. These two are the trap — 12:30 am is 00:30, but 12:30 pm is 12:30.

Adding time — bridge in stages, never add like ordinary numbers:

14:35 + 3 h 48 min → add hours: 17:35 → add up to the hour: +25 min → 18:00 (25 of the 48 used) → add the rest: 48 − 25 = 23 → 18:23

Fractional hours (for speed questions): 15 min = 0.25 h · 20 min = ⅓ h · 30 min = 0.5 h · 45 min = 0.75 h.

45 minutes is 0.75 hours, never 0.45. This single error costs more marks than any other in speed questions.

Month lengths: 30 days hath September, April, June and November; all the rest have 31, except February with 28 (29 in a leap year).

6. 2D shapes

* when the polygon is regular (all sides and angles equal)

The formula, so you never need the table: interior angles add to (n − 2) × 180°. Divide by n for each angle in a regular polygon. The shortcut: exterior angles of ANY polygon always add to 360°. So each exterior angle of a regular n-gon is 360 ÷ n, and interior = 180 − exterior. This is usually the faster route.

Triangles

Quadrilaterals

The parallelogram having zero lines of symmetry surprises nearly every child. It has rotational symmetry of order 2, which is a different thing.

Area and perimeter

7. 3D shapes — faces, edges, vertices

Euler's rule: for any solid with flat faces, Faces + Vertices − Edges = 2. Always. Use it to check your counting, or to find a missing number in the exam.

Curved solids don't obey it (they're the exception examiners like):

Vocabulary that must be exact:

• Face = a flat (or curved) surface
• Edge = where two faces meet (a line)
• Vertex = where edges meet (a corner). Plural: vertices

Prism vs pyramid: a prism has the same cross-section all the way through (two identical ends); a pyramid rises from one base to a single point (apex).

Volume and surface area

Working backwards is the exam's favourite: "a cube has surface area 294 cm² — what's its volume?" → one face = 294 ÷ 6 = 49 → side = √49 = 7 → volume = 7³ = 343 cm³.

10. Venn diagrams and sets

The two-circle picture: left-only, overlap (both), right-only, and outside (neither).

The rule that solves almost every 11+ Venn question:

Total = A + B − Both (+ Neither, if some are outside)

Because everyone in the overlap got counted twice — once in A and once in B — so you subtract them back once.

Worked example: 30 pupils; 18 play football; 15 play tennis; everyone plays at least one. How many play both?

30 = 18 + 15 − Both → Both = 33 − 30 = 3

Then fill the picture: football only = 18 − 3 = 15 · both = 3 · tennis only = 15 − 3 = 12 · check: 15 + 3 + 12 = 30 ✓

With a "neither" group: 32 children; 20 like apples; 17 like bananas; 4 like neither.

Children liking at least one = 32 − 4 = 28 → 28 = 20 + 17 − Both → Both = 9

Reading the regions — the wording that trips children:

"How many play football?" means all football players (15 + 3 = 18). "How many play only football?" means 15. Underline "only".

Carroll diagrams (the 2×2 grid version) work the same way — every child sits in exactly one of four boxes, and the four boxes add to the total.