牛津国际 AQA · International GCSE · 数学

OxfordAQA International GCSE Mathematics (9260) unit 3.1 (specification for first assessment 2018). Core rows N1–N20 run from ordering and the four operations through factors and indices, standard form, sets, rounding, FDP operators, percentage change and ratio; the Extension additions add index-form factorisation, fractional powers, exact surd work, bounds calculations, recurring decimals, reverse percentages, the compound interest formula and exponential growth and decay. The 2016 specimen pair (1E/2E, Extension tier) samples this unit at HCF, surds, fraction chains, bounds and exponential models; lines the specimens never sample are taught here as specification originals and labelled so.

Must-know for unit 3.1 (NO formulae sheet in 9260 — the recall list is the specification itself):

Order positive and negative integers, decimals and fractions on the number line with $=$, $\ne$, $<$, $>$ — convert to one form first when comparing ($\tfrac35 = 0.6$). The four operations with formal written methods cover integers, decimals, simple fractions proper and improper, and mixed numbers: $2\tfrac13 + 1\tfrac14 = \tfrac{28}{12} + \tfrac{15}{12} = \tfrac{43}{12} = 3\tfrac7{12}$. Inverse operations cancel and simplify: multiplication undoes division, subtraction undoes addition — use cancellation early ($\tfrac{15}{28} \times \tfrac{7}{9} = \tfrac{15}{4} \times \tfrac{1}{9} = \tfrac{5}{12}$ after cancelling the 7 and 3).

Even, odd, prime, factor, multiple, common factor, common multiple — with the highest common factor 最大公因数 and lowest common multiple 最小公倍数 found from prime factorisations. The Extension addition writes every factor in index form: $504 = 2^3 \times 3^2 \times 7$ and $588 = 2^2 \times 3 \times 7^2$, so the HCF is $2^2 \times 3 \times 7 = 84$ (the 1E specimen's answer) and the LCM is $2^3 \times 3^2 \times 7^2 = 3528$.

Positive integer powers with square and cube roots ($\sqrt{144} = 12$, $\sqrt[3]{125} = 5$); the index laws multiply and divide by adding and subtracting indices. The Extension addition brings fractional powers 分数指数 ($27^{1/3} = 3$, $8^{2/3} = 4$: cube-root then square) and exact calculation with surds 根式 — simplify by extracting squares and rationalise denominators with conjugates:

The 1E specimen's surd answer $4 + 4\sqrt2$ comes from this same machinery (family-level citation).

Standard form $A \times 10^n$ with $1 \le A < 10$ covers very large and very small measurements; the calculator's $\times 10^x$ key keeps arithmetic exact. Use the calculator "effectively and efficiently": know the trigonometric and square-root functions, keep full precision through the working, and round only at the end.

Sets 集合: the language and notation — $n(A)$ for the count, $A'$ the complement, $A \cup B$ the union, $A \cap B$ the intersection — with Venn diagrams to organise counts (the same toolkit as the probability unit).

Round to an appropriate degree of accuracy — decimal places and significant figures — and know each rounded value carries an interval: to the nearest unit $u$, the value lies within $\tfrac u2$ either side. The Extension addition CALCULATES with upper and lower bounds — the upper bound 上界 and lower bound 下界: choose the extreme the question needs.

Worked example (2E specimen family, adapted). A floor of 96 m² is measured to 2 s.f.; a safe occupancy needs 5.1 m² per person to 1 d.p. The LEAST number of people that must fit: divide the smallest floor by the largest space — LB $= 95.5$, UB per person $= 5.15$, so $\tfrac{95.5}{5.15} = 18.5\ldots$; you cannot have half a person and everyone must fit, so at least 19 people (the specimen's answer). State which bound each number needs BEFORE dividing.

Equivalent fractions cancel; fractions, decimals and percentages all act as OPERATORS: $35\%$ of 240 is $0.35 \times 240 = 84$; $\tfrac35$ of a quantity multiplies by $\tfrac35$ and divides by 5. Express one quantity as a fraction or percentage of another: 7 as a percentage of 28 is $\tfrac{7}{28} \times 100 = 25\%$. The Extension addition converts between fractions and recurring decimals 循环小数 in both directions — $0.\overline{45} = \tfrac{45}{99} = \tfrac5{11}$ by the algebraic shift-and-subtract method (specification line, not sampled by the specimens).

Percentage change uses multipliers: increase by $12\%$ is $\times 1.12$; decrease by $17\%$ is $\times 0.83$. The Extension additions cover reverse percentage (percentages) problems and the compound interest formula 复利公式 $A = P\left(1 + \tfrac{r}{100}\right)^n$: £2000 at 3% for 4 years gives $2000 \times 1.03^4 = £2251.02$; reverse: a £92 price after a 15% increase means an original of $92 \div 1.15 = £80$ (specification lines, not sampled by the specimens).

Ratio notation simplifies by dividing through and links to fractions (share £350 in $2:5$: parts of $\tfrac27$ and $\tfrac57$ — £100 and £250); apply ratio to problems (recipes, maps, mixtures) and use common rates 比率 including rates of pay (£9.20 per hour for 7.5 hours = £69) and best-buy comparisons (unit pricing — specification lines, not sampled). Direct proportion scales both quantities ($y = kx$); inverse proportion scales them oppositely ($y = \tfrac kx$).

The final Extension addition models exponential 指数 growth and decay: $y = a b^t$ with $b > 1$ growing and $0 < b < 1$ decaying. The 2E specimen fits a decay model from data — successive values $240, 192, 153.6$ have ratio $0.8$, so $y = 240 \times 0.8^t$ (the specimen's fitted $a$-family) — and SOLVES with it: £12000 losing 20% per year, $V = 12000 \times 0.8^t$, first falls below £2500 when $0.8^t < 0.2083$, i.e. $t > \ln 0.2083/\ln 0.8 \approx 7.0$: during the 8th year (the specimen's "how many years" answer family). Growth works identically: £1200 at 7% passes £2000 after $\ln(5/3)/\ln 1.07 \approx 7.6$, i.e. 8 full years.

• Order integers, decimals and fractions with $=$, $\ne$, $<$, $>$ on the number line (N1).
• Four operations with formal methods on integers, decimals, fractions, mixed numbers; inverse-operation cancellation (N2, N3).
• Even/odd/prime/factors/multiples vocabulary; HCF and LCM from index-form prime factorisations (N4, Extension; 1E specimen HCF 84).
• Integer powers and roots; index laws; fractional powers; exact surd arithmetic incl. rationalising (N5–N7, Extension).
• Standard form; effective calculator use (N8, N10).
• Sets: $n(A)$, $A'$, $\cup$, $\cap$ with Venn diagrams (N9).
• Rounding to appropriate accuracy; upper/lower bound calculations choosing the right extremes (N11, Extension; 2E specimen 19).
• Equivalent fractions; FDP as operators; fraction/percentage of; recurring decimal conversions (N12–N14, Extension).
• Percentage change with multipliers; reverse percentage; compound interest formula (N15, Extension).
• Ratio notation and sharing; ratio problems; rates of pay and unit pricing (N16–N19).
• Direct and inverse proportion; exponential growth and decay models fitted and solved (N20, Extension; 2E specimen family).

Boundary notes: NO formulae sheet is provided in 9260 — the recall list above is examinable as memory. Lines marked "not sampled" (fractional powers, recurring decimals, reverse percentage, rates of pay and best buys) are taught from the specification text as originals; the specimen pair is Extension-tier only, so Core-tier demand (papers 1–2 Core, 1 h 30, 80 marks) has NO local evidence and holds to spec-text teaching only. Worked examples are adapted from the 2016 specimen pair or original.

OxfordAQA 9260 unit 3.2 (specification for first assessment 2018) — the largest unit at 35% of each Extension paper. Core rows A1–A26 cover notation through graphs, equations, inequalities and sequences; the Extension additions add subject-twice rearrangement, triple expansions, all quadratic factorisation, fractional powers, quadratic algebraic fractions, formal proof, the full functions toolkit (domain, range, inverse, composite), line equations, exponential and trigonometric graphs, differentiation with tangents and stationary points, completing the square, gradients and areas under graphs, variation, every simultaneity pair, quadratic inequalities and quadratic sequences. The 2E specimen samples this unit heavily (algebra is its spine); the unsampled lines are labelled specification originals.

Must-know for unit 3.2 (NO formulae sheet):

Letters express generalised number; substitute numbers into formulae and expressions (positive, negative, fractional). The Extension addition rearranges complex formulae including when the subject appears twice: from $ax + 3 = 5(xt) - 2a$, collect the subject's terms first — $ax + 2a = 5xt - 3$, so $a(x + 2) = 5xt - 3$ and $a = \dfrac{5xt - 3}{x + 2}$. Check every rearrangement numerically.

Collect like terms; expand single brackets and (Extension) products of two or three binomials: $(x + 2)(x - 3)(x + 1) = (x^2 - x - 6)(x + 1) = x^3 - 7x - 6$. Factorise by common factors and (Extension) every quadratic $ax^2 + bx + c$, including the difference of two squares: $2x^2 - 7x - 15 = (2x + 3)(x - 5)$ and $9x^2 - 25 = (3x - 5)(3x + 5)$. Index laws run to fractional powers: $4c^5 \div 3d^3$ stays as $\tfrac{4c^5}{3d^3}$ in index-form answers (the 1E specimen's answer shape).

Rational expressions (Extension) add, subtract, multiply and divide linear or quadratic algebraic fractions — factor first, cancel whole factors, common denominators. The 2E specimen's $\dfrac{11m - 14}{m^2 - 3m - 4}$ and $\dfrac{2x + 3}{x + 4}$ answers come from this machinery.

Argue mathematically that identities hold for every value — with the Extension demand for formal proofs 证明. Structure: state the algebra, simplify, compare with the claim. Classic (2E specimen family): the expression $n^2 + n + 9$ for consecutive sums — show $n(n+1) + 9$ always sits 9 above a product of consecutive integers. A parity proof: $(2n + 1)^2 - (2n - 1)^2 = (4n^2 + 4n + 1) - (4n^2 - 4n + 1) = 8n$ — a multiple of 8 for every integer $n$. Numeric examples are never a proof.

A function maps each input to exactly one output (Core: simple input–output machines; Extension: the formal definition, $\mathrm{f}(x) = \ldots$ notation, domain 定义域 and range 值域). Composite: $\mathrm{fg}(x) = \mathrm{f}(g(x))$, apply $g$ first — the 2E specimen builds $\mathrm{fg}(x) = (x - 5)^2 + 1$ and solves $\mathrm{fg}(x) = 5$: $(x - 5)^2 = 4$ gives $x = 3$ (or $x = 7$; check: $(3 - 5)^2 + 1 = 5$). Inverse: swap and rearrange — the 1E specimen's $\mathrm{f}^{-1}(x) = \tfrac x3$ comes from $\mathrm{f}(x) = 3x$.

Coordinates in all four quadrants; plot $y = mx + c$; the Extension additions find the equation through two points or through one point with a given gradient, and use $y - y_1 = m(x - x_1)$: through $(1, 7)$ and $(2, 9)$, $m = 2$ and $y = 2x + 5$. Parallel lines share $m$; perpendicular gradients multiply to $-1$ (specification line, not sampled).

Sketch and interpret linear, quadratic, cubic and reciprocal graphs; the Extension additions add exponential functions $y = k^x$ for positive $k$ and the trigonometric graphs (degrees) — both specification lines not sampled by the specimens. Identify roots, intercepts and turning points; the Extension addition DEDUCES turning points by completing the square 配方: $y = x^2 - 6x + 5 = (x - 3)^2 - 4$ has its minimum at $(3, -4)$ (specification line, not sampled). Gradients and areas under graphs (including quadratic and non-linear graphs) are estimated by tangents and trapezia, and — at Extension — computed exactly by differentiation (next section).

The Extension calculus block: understand and use the gradient function 导函数 $\dfrac{dy}{dx}$, differentiate $kx^n$ for positive integer $n$ (and sums), and know that the gradient of a function at a point IS the gradient of the tangent 切线 there — then work out the tangent's equation.

The 2E specimen's values (family citation): a curve with $\dfrac{dy}{dx} = 15$ at the point carries tangent $y = 15x - 36$. Stationary points solve $\dfrac{dy}{dx} = 0$, classified as maxima, minima 极值 or points of inflection 拐点, with curves sketched from the derivatives:

Worked example (original). $y = x^3 - 12x + 5$: $\dfrac{dy}{dx} = 3x^2 - 12 = 0$ gives $x = \pm 2$. At $x = -2$, $y = 21$ (gradient changes $+$ to $-$: maximum); at $x = 2$, $y = -11$ (changes $-$ to $+$: minimum). At $x = 3$ the gradient is $3(9) - 12 = 15$ and the tangent is $y + 4 = 15(x - 3)$, i.e. $y = 15x - 49$.

Solve linear equations (unknown both sides), quadratics by factorising and (Extension) by completing the square and the quadratic formula — the 1E specimen's $x = -3 \pm \sqrt7$ comes from $x^2 + 6x + 2 = 0$: $x = \tfrac{-6 \pm \sqrt{36 - 8}}{2}$. Simultaneous pairs: two linear equations, and (Extension) one linear with one quadratic 二次方程. Variation (Extension): direct $y = kx^n$ and inverse $y = \tfrac{k}{x^n}$ in algebraic terms (specification line, not sampled). Inequalities: linear in one variable, and (Extension) in one or two variables plus quadratic inequalities 二次不等式 — solve $x^2 - x - 6 > 0$ by critical values $x = -2, 3$: positive quadratic, so $x < -2$ or $x > 3$ (specification line, not sampled).

Generate terms from term-to-term and position-to-term rules; use triangular, square and cube number sequences (Core) with the Extension additions covering quadratic sequences 二次数列: second differences are constant, and the nth term is $an^2 + bn + c$ with $a = \tfrac12(\text{second difference})$. For $5, 9, 17, 29, 45, \ldots$: first differences $4, 8, 12, 16$, second difference $4$, so $a = 2$. Take $2n^2$ (which gives $2, 8, 18, 32, 50$) and subtract: the residuals $3, 1, -1, -3, -5$ are linear, $-2n + 5$. So the nth term is $2n^2 - 2n + 5$ — check $n = 4$: $32 - 8 + 5 = 29$. ✓

• Generalised notation; substitution; rearrangement with the subject twice (A1–A3, Extension).
• Expanding two or three binomials; factorising all quadratics and the difference of two squares; fractional index answers; quadratic algebraic fractions (A4–A7, Extension; 1E/2E specimens).
• Formal algebraic proof (A8, Extension; 2E n² + n + 9 family).
• Functions: definition, domain, range, composite fg, inverse (A9, Extension; 2E fg(x) = (x−5)² + 1, 1E f⁻¹(x) = x/3).
• Coordinates; line equations through two points or point + gradient; perpendicular gradients (A10–A11; the latter not sampled).
• Linear, quadratic, cubic, reciprocal, exponential and trigonometric graphs; turning points by completing the square; gradient/area estimates (A12, A16–A17; several lines not sampled).
• Differentiation of kxⁿ; gradient = tangent gradient; tangent equations; stationary points classified with sketches (A13–A15, Extension; 2E tangent family).
• Linear equations; quadratics by all three methods; linear–linear and linear–quadratic simultaneity (A19–A21, Extension).
• Context into algebra (A22, Extension); linear (one/two variable) and quadratic inequalities with solution sets (A23, Extension; not sampled).
• Sequence types; nth terms linear and quadratic (A24–A26, Extension).

Boundary notes: NO formulae sheet in 9260. Lines marked "not sampled" (perpendicular gradients, exponential/trig sketches, completing-the-square turning points, variation, quadratic and two-variable inequalities) are specification originals; the specimens are Extension-tier only, so Core-tier demand has no local evidence. Worked examples are adapted from the 2016 specimen pair or original.

OxfordAQA 9260 unit 3.3 (specification Version 5.1) — at 31.5% of the combined specimen marks. Core rows G1–G26 cover angle facts through constructions, mensuration, similarity, trigonometry, transformations, vectors and matrices; the Extension additions add the full circle-theorem programme with proof, congruence conditions, bearings, sphere/pyramid/cone/frustum formulae, k-k²-k³ similarity, sectors, 3D Pythagoras and trigonometry, the sine/cosine rules with ½ab sinC, combined transformations with fractional/negative scale factors, vector geometry, and the 2×2 matrix programme (identity, unit-square transformations, combinations). The 1E specimen is the geometry paper; unsampled lines are labelled specification originals.

Must-know for unit 3.3 (NO formulae sheet):

Conventional terms: points, lines, vertices, edges, planes; parallel, perpendicular; polygons and regular polygons; the standard triangle labelling. Angle facts: at a point (360°), on a line (180°), vertically opposite; parallel lines give alternate 内错角, corresponding 同位角 and interior angles (the "co-located" names, never "Z angles"); triangle sum 180° with the isosceles, equilateral and scalene families. Polygon sums: interior $(n-2) \times 180°$, exterior $360°$; for regular polygons also the angle at the centre $= \tfrac{360°} {n}$. The special quadrilaterals (square, rectangle, parallelogram, trapezium, kite, rhombus) carry their side/angle/diagonal properties, and 2D shapes have reflection and rotation symmetry counts (a regular hexagon: 6 lines, order 6).

Congruence 全等 with the Extension conditions for triangles — SSS, SAS, ASA, RHS — each justified by which three independent measurements pin a triangle. Similarity 相似 scales lengths by $k$: (Extension) areas by $k^2$, volumes by $k^3$, in every direction.

The Extension circle programme: angle at the centre is twice the angle at the circumference; the semicircle angle is 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; tangent $\perp$ radius; tangents from an external point are equal; the perpendicular from the centre bisects a chord; the alternate segment theorem. The verified chain: an angle at the circumference of 35° subtends the same arc as a centre angle of 70°, and the angle from the OPPOSITE arc is $\tfrac12(360° - 70°) = 145°$ — the two circumference angles are opposite angles of a cyclic quadrilateral ($35° + 145° = 180°$) — every step names its theorem (the 1E specimen's circle question carries answers of this family; G9's geometric proof 证明 demand).

Faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres; plans and elevations (front, side, top views — a specification line not sampled by the specimens). Measure line segments and angles; scale drawings with scale factors; bearings 方位角: three-figure, clockwise from north (Extension includes the eight compass points) — the 2E specimen's 252° comes from a bearings chain: a return bearing plus an angle at a point.

Ruler-and-compass constructions 作图: perpendicular bisector of a segment, perpendicular to a line from/at a point, angle bisector, 60° angle — used to build figures and solve loci 轨迹 problems; the perpendicular distance from a point to a line is the shortest. Leave all arcs visible: the arcs are the working.

Standard units with conversions (time 12/24-hour; length, area, volume/capacity, mass; compound units speed and density). Areas: triangle $\tfrac12 bh$, parallelogram $bh$, trapezium $\tfrac12(a+b)h$; volume by $V = Ah$ (constant cross-section). The circle: $2\pi r$, $\pi r^2$; (Extension) sectors 扇形 arc $\tfrac\theta{360} \times 2\pi r$ and area $\tfrac\theta{360} \times \pi r^2$, plus sphere/pyramid/cone surface and volume formulae and composite solids including frustums (specification lines, several not sampled) — a cone of $r = 5$, $h = 12$ has slant 13, $V = 100\pi$ and curved surface $65\pi$; the 1E specimen's cone problem finds a proportion $p = 15$. Similar figures (Extension): lengths $k$ : areas $k^2$ : volumes $k^3$ — the 2E specimen's similar solids give 864 cm³ from a volume ratio.

$a^2 + b^2 = c^2$ with SOH CAH TOA in right-angled triangles — 2D always, 3D figures at Extension: extract the right triangle first (the 1E specimen's 3D trig gives 62.7°). The sine rule $\dfrac a{\sin A} = \dfrac b{\sin B}$ and cosine rule $a^2 = b^2 + c^2 - 2bc\cos A$ cover any triangle (Extension), with Area $= \tfrac12 ab\sin C$ reversible — the 2E specimen's cosine rule gives 102.6°, and $\cos x = \tfrac8{11}$ yields $x = 43.3°$.

Worked example (2E cosine-rule family, adapted). A triangle has $a = 9$, $b = 11$ and angle $C = 72°$. $c^2 = 81 + 121 - 2(9)(11)\cos 72° = 202 - 198 \times 0.309 = 202 - 61.2 = 140.8$, so $c = 11.9$ (3 s.f.). Then the smallest angle sits opposite the shortest side: $\cos A = \tfrac{b^2 + c^2 - a^2}{2bc} = \tfrac{121 + 140.8 - 81}{2 \times 11 \times 11.87} = \tfrac{180.8}{261.1} = 0.693$, so $A = 46.2°$ (keeping $c$ unrounded) — check $A + B + C = 180°$ with $B = 61.8°$. ✓

Single rotations, reflections, translations (by a vector) and enlargements (positive factor) — with the preserved properties named; the Extension additions cover combined transformations and enlargements by fractional and negative scale factors 负比例因子 (an inverted, shrunken image on the opposite side of the centre). Vectors 向量: notation, sum, difference and scalar multiples (graphically and by components), the commutative and associative laws, and simple 2D geometry — the 2E specimen proves $CB \parallel MN$ via a $\tfrac12$ ratio using vectors.

The 2×2 matrix 矩阵 programme (Extension): multiplication (2×2 by 2×2, by 2×1, scalar), the identity matrix 单位矩阵 $I$; transformations of the UNIT SQUARE — rotations of 90°/180°/270° about the origin, reflections in $x = 0$, $y = 0$, $y = x$, $y = -x$, enlargements centred on the origin — represented by matrices such as $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$ (reflection in $y = -x$, the 1E specimen's matrix); combined transformations by matrix multiplication, in the order the transformations apply.

Worked example (1E matrix-solve family, adapted). Solve $\begin{pmatrix} 3 & 1 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 10 \\ 5 \end{pmatrix}$: the equations are $3x + y = 10$ and $x + 2y = 5$. From the first, $y = 10 - 3x$; substituting: $x + 2(10 - 3x) = 5$, so $-5x = -15$, giving $x = 3$ and $y = 1$. Check: $3(3) + 1 = 10$. ✓ The 1E specimen's own pairs $(6, -\tfrac12)$ and $(-2, -\tfrac52)$ come from the same method on a second, linked system.

• Conventional terms and triangle notation; angle facts; parallel-line angles; polygon interior/exterior/centre sums; quadrilateral properties; symmetry counts (G1–G6).
• Congruence with SSS/SAS/ASA/RHS; similarity with k, k², k³ ratios (G7, G17, Extension).
• Circle language and the full theorem bank with proof-style reasons (G8–G9, Extension; 35° → 70° → 145° verified chain).
• 3D shape properties; plans and elevations (not sampled); measuring, scale factors, bearings (G10–G12).
• Constructions (four ruler-and-compass types) with loci; shortest distance property (G13).
• Units and compound measures; area and V = Ah formulae; circle formulae; sectors; sphere/pyramid/cone/frustum surface and volume; composite shapes (G14–G16, Extension).
• Pythagoras and trigonometry in 2D and 3D; sine/cosine rules and ½ab sinC (G19–G20, Extension; 1E 62.7°, 2E 102.6° and cos x = 8/11).
• Transformations including combined and fractional/negative enlargements (G21, Extension).
• Vector notation and geometry (G22; 2E CB ∥ MN proof).
• Matrix multiplication, identity, unit-square transformations and combinations (G23–G26, Extension; 1E matrices).

Boundary notes: NO formulae sheet in 9260. Lines marked "not sampled" (plans and elevations, parts of the solids programme) are specification originals; the specimens are Extension-tier only. Worked examples are adapted from the 2016 specimen pair or original.

OxfordAQA 9260 unit 3.4 (specification Version 5.1) — the smallest unit at 13% of the combined specimen marks. Core rows S1–S5 cover data types, reading tables, two-way tables, the full chart bank and the basic averages; the Extension additions add histograms with UNEQUAL class intervals, cumulative frequency diagrams, box plots, and quartiles with the inter-quartile range and percentiles. The 1E specimen carries a histogram and a cumulative-frequency correction to upper boundaries; the 2E specimen's statistics questions include the 40-men and 38% reads. Lines the specimens never sample are labelled specification originals.

Must-know for unit 3.4:

Qualitative 定性 data categorises (favourite fruit); discrete 离散 data counts (number of goals); continuous 连续 data measures (height), grouped or ungrouped. Extract data from printed tables and lists — read the row and column headers first, and quote units. Two-way tables 双向表 cross-tabulate two variables; every row and column total checks the interior cells: 30 students, 18 cricket, 15 tennis and 7 both fills a 2×2 whose cells must sum to 30.

The Core chart bank: scatter graphs, stem-and-leaf, tally charts, pictograms, bar charts, dual and composite bar charts, pie charts, line graphs, frequency polygons and histograms with EQUAL class intervals. Choose by data type — categories take bars, comparisons take dual bars, parts-of-a-whole take pie charts, trends take line graphs, correlation takes scatter graphs.

A histogram 直方图 with unequal class intervals 不等组距 draws bar HEIGHT as the frequency density 频率密度 $=$ frequency over class width — the AREA is the frequency (classes of widths 2, 3, 5, 10, 15 holding 5, 12, 18, 14, 9 give heights 2.5, 4, 3.6, 1.4, 0.6). A cumulative frequency diagram 累积频率图 plots each running total at its UPPER class boundary as a smooth increasing curve (the 1E specimen's correction is exactly this upper-boundary point placement); the median reads at $\tfrac n2$, quartiles at $\tfrac n4$ and $\tfrac{3n}4$. A box plot 箱线图 (five-number summary: minimum, $Q_1$, median, $Q_3$, maximum) shows spread at a glance — the box spans the IQR, the whiskers the range, outliers by judgement.

The mean 平均数 divides the total by the count (grouped data multiplies each MIDPOINT by its frequency first); the median 中位数 orders and takes the middle; the mode 众数 is most frequent and the modal class 众数组 is the class with the highest frequency DENSITY when widths differ. The Extension additions add quartiles 四分位数 and the inter-quartile range 四分位距 $= Q_3 - Q_1$ (the median of the lower and upper halves), and percentiles 百分位数 read like the median at any percentage — the $p$th percentile is the value $\tfrac p{100}n$ of the way up the ordered data or the cumulative-frequency curve.

Worked example (original). 60 workers' times have cumulative frequencies 6, 17, 27, 42, 60 at upper boundaries 10, 15, 20, 25, 30 (hours). The median reads at worker 30: between 20 (27) and 25 (42), so $20 + 5 \times \tfrac{30 - 27}{42 - 27} = 21$ hours. $Q_1$ at worker 15: $10 + 5 \times \tfrac{15 - 6}{17 - 6} = 14.1$; $Q_3$ at worker 45: $25 + 5 \times \tfrac{45 - 42}{60 - 42} = 25.8$; IQR $\approx 11.7$ hours. Draw the reading lines — the marks are for the reads.

• Data types qualitative/discrete/continuous; grouped and ungrouped (S1).
• Extract from printed tables and lists, quoting units (S2).
• Design and use two-way tables with row/column checks (S3).
• The Core chart bank, chosen by data type (S4).
• Unequal-interval histograms by frequency density (area is frequency); cumulative frequency diagrams at UPPER boundaries with median/quartile reads; box plots (S4 Extension; 1E histogram and CF-boundary anchors).
• Mean (grouped via midpoints), median, mode and modal class (S5).
• Quartiles, inter-quartile range and percentiles from data or the curve (S5 Extension).

Boundary notes: probability vocabulary sits in this specification's Number/Algebra work and earlier units' sets; the AO split is asserted by the specification. Lines not sampled by the specimens (stem-and-leaf, pie charts and several chart types on the Extension papers) are taught as specification originals; the specimens are Extension-tier only, so Core-tier demand has no local evidence. Worked examples are adapted from the 2016 specimen pair or original.