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1. 数论

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1. 数论

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3.1 · 数论

Exact arithmetic and estimation: A prime number that divides the integer exactly.
Indices, surds and standard form: The power to which a base is raised.
Accuracy, bounds and compound measures: The smallest possible value consistent with a stated rounding rule.
Signed numbers, place value and operation order: A number that multiplies a nonzero original number to give 1.
Exact fractions and mixed-number operations: A fraction whose numerator is at least as large as its positive denominator.
Systematic lists and the product rule: A list made in a fixed order so every allowed outcome appears exactly once.
Exact surds and rationalising denominators: An irrational root kept in exact form rather than replaced by a rounded decimal.
Terminating and recurring decimal conversions: A decimal with a digit or block of digits that repeats forever.
Metric and compound-unit conversions: The multiplier that expresses the same quantity in another unit.
Estimation, rounding and simple error intervals: A digit counted from the first nonzero digit when reporting numerical precision.
Ratio shares and fraction operators: A comparison that states the relative numbers of equal-sized parts.

质因数: 能整除该整数的质数。
索引: 底数被提升的幂次。
下界: 符合给定舍入规则的最小可能值。
倒数: 一个将非零原数相乘得到1的数。
假分数: 分子至少等于其正分母的分数。
系统列表: 按固定顺序排列的列表，使每个允许的结果恰好出现一次。
根式: 保留精确形式的无理根，而非替换为四舍五入的小数。
循环小数: 具有无限重复的数字或数字块的小数。
换算因子: 用于在同一数量用另一单位表示时的换算系数。
有效数字: 报告数值精度时从第一个非零数字开始计算的位数。
比值: 陈述等大小部分相对数量的比较。

3.2 · 代数

Equations, identities and rearrangement: An equality that holds for every allowed value of its variable.
Quadratics and inequalities: The quantity b²-4ac that determines the real roots of ax²+bx+c=0.
Simultaneous equations and feasible regions: Combining equations to remove one variable while retaining the same solutions.
Coordinate geometry and tangents: The change in y divided by the corresponding change in x.
Sequences, series and recurrence: The constant multiplier between consecutive terms of a geometric sequence.
Algebraic notation, substitution and vocabulary: The numerical factor multiplying a variable term.
Collecting, expanding and factorising expressions: Rewrite an expression as a product of factors.
Rearranging formulae and checking the subject: The quantity written alone on one side of a formula.
Identities, equivalence and algebraic arguments: An expression with the same value as another for every allowed input.
Function machines and reversing operations: The value supplied to a rule before its operations are carried out.
Domains, inverses and composition: The set of allowed inputs to a function.
Coordinate quadrants and basic graph families: A line approached by a graph without a finite crossing in the stated example.
Exponential and degree-based trigonometric graphs: The input interval after which a repeating graph has the same values again.
Translations and reflections of function graphs: Move every point of a graph by the same horizontal and vertical displacement.
Distance-time graphs and contextual intersections: Not changing position over the time interval described.
Graphical gradients and area estimates: The slope of a tangent, used to estimate a curve’s rate at one input.
Origin-centred circles and tangent equations: The slope of a radius joining the centre to a point on the circle.
Numerical iteration for equation roots: Repeat a rule, using the previous output as the next input.
Linear inequalities and number-line solutions: All inputs that satisfy the stated condition.
Quadratic inequalities and two-variable regions: The equality line separating values that satisfy an inequality from those that do not.
Figurate, geometric and Fibonacci-type sequences: A rule that generates new terms from preceding terms.
Quadratic nth terms and surd-ratio progressions: The difference between consecutive first differences of a sequence.
Algebraic fractions and excluded values: An input for which an original denominator or divisor is zero.

身份认同: 对其变量的每一个允许值都成立的等式。
判别式: b²-4ac 是决定 ax²+bx+c=0 实数根的判别式。
消除反应: 合并方程以消去一个变量同时保持解集不变。
斜率: y的变化量除以相应的x变化量。
公比: 等比数列中相邻两项之间的常数乘数。
系数: 乘于变量项前的数值系数。
因式分解: 将表达式改写为因式的乘积形式。
主题: 公式一侧单独写出的量。
等价表达式: 对于所有允许输入值均具有相同值的表达式。
输入: 在规则执行运算之前提供给规则的⟦number⟧值。
定义域: 函数的允许输入集合。
渐近线: 在给定示例中，图形趋近但无有限交点的一条直线。
周期: 重复图形再次出现相同数值之前的输入区间。
图像平移: 将图形的每个点按相同的水平和垂直位移进行移动。
静止: 在描述的时间间隔内位置不发生改变。
切线斜率: 切线的斜率，用于估算曲线在某一个输入处的变化率。
半径斜率: 连接圆心与圆上某点的半径的斜率。
迭代的示例: 重复应用规则，将前一次的输出作为下一次的输入。
解集: 所有满足所给条件的输入。
边界线: 分隔满足不等式与不满足不等式的数值的等号线。
递归规则: 利用先前项生成新项的规则。
二阶差分: 数列连续一阶差分之间的差。
使分母为零的值: 使原始分母或除数为零的输入。

3.3 · 比率、比例及变化率

Percentages, ratio and proportional reasoning: A factor that performs a percentage change in one multiplication.
Regression, financial models and residuals: The observed value minus the value predicted by a fitted model.
Ratios, equivalent proportions and mixtures: A comparison of one share with the combined quantity.
Scale drawings and maps: The ratio of a drawing length to its corresponding real length.
Direct and inverse proportional relationships: The fixed multiplier in a stated proportional relationship.
Rates, unit prices, density and pressure: Mass per unit volume, with both units stated.
Length, area and volume scale factors: The squared length factor between similar shapes.
Simple interest, compound growth and decay: Interest calculated on the updated balance each period.
Iterative processes with repeated deposits: A rule obtaining a later value from the preceding value.

乘数: 在一次乘法运算中实现百分比变化的因子。
剩余: 观测值减去拟合模型预测的值。
部分与整体之比: 单份与组合数量的比较。
规模: 绘图长度与其对应实际长度的比值。
比例常数: 明确比例关系中的固定乘数。
密度: 单位体积的质量，同时注明两个单位。
面积比例因子: 相似图形之间的平方长度因子。
复利: 每个周期基于更新后的余额计算的利息。
递推关系式: 从前一项推导后一项的规则。

3.4 · 几何与度量

Angle reasoning, similarity and mensuration: The multiplier that relates corresponding lengths in similar shapes.
Right triangles and non-right triangles: The side opposite the right angle in a right-angled triangle.
Vectors and transformation geometry: The vector sum representing the combined displacement or force.
Reflections, rotations and enlargements: The point from which each point's displacement is multiplied by a scale factor.
Constructions, loci and geometric conditions: The set of all points satisfying a stated geometric condition.
Circle theorems and reasoned proofs: A quadrilateral whose four vertices lie on one circle.
Shape vocabulary, properties and symmetry: A mirror line that maps a whole shape onto itself.
Angle facts and polygon reasoning: Angles in matching positions at a transversal crossing two lines.
Triangle congruence and geometric proof: Equal shape and size, with matching lengths and angles.
Circle parts and geometric definitions: A region bounded by a chord and its corresponding arc.
Geometric reasoning on coordinate axes: The point halfway between the endpoints of a segment.
Solid properties, plans and elevations: An orthographic view from a stated horizontal direction.
Measuring angles and three-figure bearings: A direction measured clockwise from north at the starting point.
Combined isometries and invariants: A property preserved by a specified transformation.
Circle angles and proof chains: An angle formed by two radii meeting at the centre.
Tangents, chords and the alternate segment: The tangent–chord angle equals the angle subtended by that chord in the opposite segment.
Areas, perimeters and composite plane shapes: The distance between a base and its parallel opposite level, measured at right angles.
Prism volume and cylinder measurement: A slice perpendicular to a prism’s length that remains constant along it.
Circle lengths, areas and composite boundaries: The total length of a circle’s boundary.
Spheres, cones, pyramids and frustums: The remaining solid after a cone or pyramid is cut parallel to its base.
Arcs, sectors and reverse angle calculations: A circle region bounded by two radii and their intervening arc.
Exact trigonometric values from special triangles: A value retained as a fraction or surd rather than a rounded decimal.
Sine rule, cosine rule and triangle area: The angle between the two named sides.
Right triangles in three-dimensional shapes: The view or component of a spatial length on a selected plane.
Vector geometry and midpoint arguments: Lying on one straight line.

缩放因子: 关联相似图形对应长度的乘数。
斜边: 直角三角形中直角所对的边。
合力: 表示合位移或合力的向量和。
放大中心: 每个点的位移按缩放因子进行乘法的原点。
基因座: 满足特定几何条件的所有点的集合。
圆内接四边形: 四个顶点共圆的四边形。
对称轴: 能将整个图形映射到其自身的一条对称轴（镜像线）。
同位角: 横截线与两直线相交时处于相同位置关系的角（同位角）。
全等: 形状和大小完全相等，对应边长和角度相匹配。
细分: 由一条弦及其对应的弧所围成的区域（弓形）。
中点: 线段两个端点正中间的点。
立面图: 从指定水平方向观察得到的正投影视图。
方位角: 从起点起按顺时针方向测量的方位角。
不变量: 经指定变换后保持不变的性质。
圆心角: 两条半径在圆心处相交所形成的角（圆心角）。
弦切角定理: 切线与弦所夹的角等于该弦在相对弓形内所对的圆周角（弦切角定理）。
垂直高度: 底面与其平行的对立面之间的距离，沿垂直方向测量。
截面: 垂直于棱柱长度方向的截面，且沿棱柱长度保持不变。
周长: 圆边界线的总长度（周长）。
圆台: 圆锥或棱锥被平行于底面的平面截去顶部后剩余的几何体。
扇形: 由两条半径及其所夹圆弧围成的圆形区域。
精确值: 以分数或根式形式保留的值，而非四舍五入的小数。
夹角: 两条指定边之间的夹角。
投影: 空间长度在选定平面上的投影或分量。
共线: 位于同一直线上。

3.5 · 概率

Probability, trees and conditional reasoning: The probability of an event after restricting the sample space to a stated condition.
Experimental probability, fairness and expected outcomes: The observed count of an outcome divided by the number of trials.
Systematic possibilities and equally likely outcomes: The complete set of outcomes in an experiment.
Venn diagrams, unions and two-way counts: The outcomes belonging to both named sets.
Conditional probability with tables and expected frequencies: A probability within the group selected by given information.

条件概率: 在给定信息所选组内的概率。
相对频率: 某结果的观测频数除以试验次数。
样本空间: 实验中所有结果的完整集合。
交点: 同时属于两个指定集合的结果。

3.6 · 统计学

Data summaries, histograms and interpretation: Frequency divided by class width, used as histogram height.
Cumulative frequency and box plots: The difference between the upper and lower quartiles.
Samples, populations and justified comparisons: The complete group about which a statistical claim is made.
Choosing displays, pie charts and time series: Measurements recorded in time order.
Frequency summaries and grouped estimates: The class interval containing the greatest frequency.
Scatter graphs, correlation and cautious predictions: A pattern of association between two measured variables.
Histogram density and cumulative-frequency construction: Frequency divided by class width, used as histogram height.
Quartiles, box plots and distribution comparisons: The upper quartile minus the lower quartile.

频率密度: 频率除以组距，用作直方图的高度。
四分位距: 上四分位数减去下四分位数。
人口: 统计声明所针对的完整群体。
时间序列: 按时间顺序记录的测量值。
众数组: 包含最大频率的组距区间。
相关性: 两个测量变量之间的关联模式。

备考指南

Choose one tier for all three papers. Foundation targets grades 1–5; Higher targets grades 4–9, with a possible grade 3 award.
The full formula sheet is supplied for 2026 and 2027; use the correct tier and year. The September 2026 notice also confirms continued provision from 2028 for this specification.

仍需教学覆盖内容

Official question-bank boundaries, scheme alignment and all objective-level teaching coverage require the recorded review; no practice registry promotion.
The authored diagnostic assessments are not full-length qualification mocks.