The 11+ reference sheet

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.

---

8. Bearings

Bearings appear in the harder GL maths papers and are worth easy marks once the three rules are automatic.

The three rules — all three, every time:

1. Measured from North
2. Measured clockwise
3. Always three digits (so 45° is written 045°)

The eight compass points

Turning questions: "Facing NE, you turn 225° clockwise. Which way are you now facing?" → 045 + 225 = 270 → West. If you go over 360, subtract 360.

Back bearings (the bearing of A from B, given the bearing of B from A): add 180 if under 180, subtract 180 if 180 or over.

The bearing of B from A is 070° → the bearing of A from B is 070 + 180 = 250° The bearing of B from A is 240° → the bearing of A from B is 240 − 180 = 060°

The wording trap: "the bearing of A from B" means stand at B and look at A. The word after "from" is where you stand. Children reliably read it backwards — underline the "from" word every time.

---

9. Algebra and equations

The vocabulary:

• Term: a single part (3x, or 7)
• Coefficient: the number in front (in 5y, the coefficient is 5)
• Expression: no equals sign (3x + 2)
• Equation: has an equals sign (3x + 2 = 14)

Collecting like terms: only terms with the same letter combine.

5a + 3b + 2a − b = 7a + 2b (a's with a's, b's with b's)

Solving — undo in reverse order. The equation did things to the letter; undo them backwards.

4b − 5 = 23 → the machine did "×4 then −5" → undo: +5 then ÷4 23 + 5 = 28 → 28 ÷ 4 → b = 7

Brackets — undo the outside first:

3(x − 2) = 18 → ÷3 → x − 2 = 6 → +2 → x = 8

Substitution: replace the letter, then compute.

y = 4x + 2, x = 5 → y = 4(5) + 2 = 22

Function machines backwards — walk back through, inverting each step:

in → ×5 → −12 → out = 43. Undo the last step first: 43 + 12 = 55, then 55 ÷ 5 = 11

The nth term: find the difference (that's the multiplier), then adjust.

6, 10, 14, 18… difference 4 → rule is "4n ± something". 4×1 = 4 but term 1 is 6, so 4n + 2. Always verify on term 2: 4(2) + 2 = 10 ✓. Then the 10th term = 4(10) + 2 = 42.

The consecutive-numbers shortcut: three consecutive numbers summing to S → the middle one is S ÷ 3.

Three consecutive odd numbers add to 87 → middle = 29 → they are 27, 29, 31.

---

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.

---

11. Things to revise — the checklist

Tick each when your child can do it without hesitating. Anything unticked two weeks before the exam is where the remaining time goes.

Number

• Place value to millions, including numbers with zeros in the middle
• Rounding to 10/100/1,000/10,000, including the carry cases (148,509 → 149,000)
• Times tables to 12 × 12 — instantly, not counted
• Long multiplication and division
• Primes under 100 + the five fake primes
• Squares to 15², cubes to 10³
• Factors, multiples, HCF, LCM
• Divisibility tests (especially 3, 4, 9)
• Negative numbers on a number line, including subtracting a negative
• BODMAS order of operations

Fractions, decimals, percentages

• The equivalence table (section 3) both directions
• Add/subtract fractions with different denominators
• Multiply and divide fractions
• Fraction of an amount; percentage of an amount via 10%
• Percentage change and reverse percentage (£24.50 after 30% off → £35)
• Simplifying to lowest terms

Ratio and proportion

• Share in a ratio (parts → one part → answer)
• Given one side, find the other
• Recipe scaling (unitary method) and best-buy
• Inverse proportion (workers/time)

Algebra

• Collecting like terms
• Solving one-step, two-step and bracket equations
• Substitution
• Function machines forwards and backwards
• nth term, with the verify-on-term-2 habit
• Sequences: constant difference, changing difference, ×, ×then+, Fibonacci

Shape and space

• Angles: straight line 180°, point 360°, triangle 180°, quadrilateral 360°
• Vertically opposite angles equal
• Interior/exterior angles of regular polygons
• Area & perimeter of rectangle, triangle, parallelogram, trapezium
• Compound (L-shape) area and perimeter — including deducing missing sides
• 3D solids: faces, edges, vertices + Euler check
• Volume of cube/cuboid; surface area; working backwards from them
• Coordinates in four quadrants; reflection and translation
• Lines of symmetry; rotational symmetry order
• Bearings: three digits, from North, clockwise; back bearings

Measurement and time

• All the unit conversions (section 4), including the m²/cm² trap
• 12-hour ↔ 24-hour, including midnight and noon
• Adding/subtracting time by bridging
• Timetables
• Speed = distance ÷ time, with fractional hours
• Average speed = total distance ÷ total time
• Roman numerals to 2026

Data and probability

• Mean, median, mode, range
• Reverse mean (totals thinking)
• Bar charts, line graphs, pictograms (read the key!), frequency tables
• Pie chart angles both directions
• Probability as a fraction; "not" questions
• Two events: multiply; two dice: ordered pairs
• Venn diagrams: Total = A + B − Both

Verbal reasoning

• The 411-word vocabulary list
• Letter codes (write the alphabet out first)
• Letter arithmetic
• Number sequences: test the five rule families in order
• Analogies: name the relationship before looking at options
• Odd one out: find what the four share
• Logic puzzles: draw the picture in the margin

Exam technique

• Two-pass method (dot the hard ones, come back)
• Never leave a blank
• Eliminate before choosing
• Check question number against answer-sheet row every 5 questions
• Re-read the instruction line at every new block