📚 Year 7 AQA Further Mathematics: Core Knowledge Summary | Year 7 AQA 进阶数学：核心知识点梳理
Year 7 Further Mathematics on the AQA pathway bridges the gap between Key Stage 3 fundamentals and the deeper reasoning required for higher-level study. This article distills the core topics — from number properties to algebraic manipulation, geometry and statistics — into clear, bilingual explanations. Use it as a revision checklist or a first-time learning guide to build confidence and accuracy.
AQA 进阶数学课程为 7 年级学生衔接了 KS3 基础与高阶思维所需的能力。本文提炼了数的性质、代数运算、几何与统计等核心主题，以中英双语呈现清晰讲解，可作为复习清单或初学指南，帮你建立信心、提升准确度。
1. Number Properties | 数的性质
Understanding factors, multiples and primes is essential. A prime number has exactly two distinct factors: 1 and itself. For example, 2, 3, 5, 7 and 11 are all primes, whereas 1 is not prime. The highest common factor (HCF) of two numbers is the largest integer that divides both without remainder, while the lowest common multiple (LCM) is the smallest positive integer that is a multiple of both.
理解因数、倍数与质数是进阶数学的基础。质数恰好有两个不同的因数：1 和它本身。例如 2、3、5、7、11 都是质数，而 1 不是。两个数的最大公因数 (HCF) 是能同时整除它们的最大整数，最小公倍数 (LCM) 则是它们共有的最小正倍数。
f 3, e.g. 2³ = 8, 3³ = 27. Index notation makes repeated multiplication compact; for instance, 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.
平方数由整数自乘得到：1² = 1，2² = 4，3² = 9，直到 10² = 100。立方数是整数的 3 次幂，如 2³ = 8，3³ = 27。指数记法使连乘更简洁，例如 2⁵ = 2 × 2 × 2 × 2 × 2 = 32。
form helps in finding HCF and LCM efficiently by comparing the index of each prime factor.
质因数分解可将任意整数表示为质数的乘积。60 可写成 2² × 3 × 5。这种形式便于比较各质因数的指数，从而快速求得 HCF 和 LCM。
2. Fractions, Decimals and Percentages | 分数、小数和百分比
Equivalence between fractions, decimals and percentages is a key skill. A fraction like 3/5 equals 0.6 and 60%. To convert a fraction to a decimal, divide the numerator by the denominator; to change a decimal to a percentage, multiply by 100. Recurring decimals such as 0.3̅ (0.333…) can be written as fractions using algebraic methods, but at this level recognising common equivalents is sufficient.
分数、小数和百分比之间的等价转换是一项关键技能。像 3/5 这样的分数等于 0.6 和 60%。将分数化为小数时用分子除以分母，将小数化为百分比时乘以 100。循环小数如 0.3̅ 可通过代数方法化为分数，但现阶段掌握常见等价关系即可。
Operations with fractions follow consistent rules. For addition and subtraction, find a common denominator; for multiplication, multiply numerators and denominators straight across; for division, multiply by the reciprocal. Always simplify the answer to its lowest terms.
分数的运算遵循固定规则。加减法时需先通分，乘法直接将分子分母分别相乘，除法则乘以倒数。最后务必把结果化为最简分数。
Percentage increase and decrease are applied frequently in real-life problems. To increase £40 by 15%, calculate 15% of £40 = £6, then add to get £46. Alternatively, use a multiplier: 1.15 × 40 = 46. For a decrease of 20%, multiply by 0.8.
百分比增减在生活中应用广泛。将 £40 增加 15%，先算出 £40 的 15% 为 £6，相加得 £46；或用倍数 1.15 × 40 = 46。减少 20% 时则乘以 0.8。
3. Algebraic Manipulation | 代数运算
b = 2a + 6b. Remember that only terms with exactly the same variable part can be combined.
代数用字母表示未知数或变量。合并同类项可简化表达式：3a + 2b − a + 4b = 2a + 6b。只有所含字母完全相同的项才能合并。
ppears outside the bracket, it changes the sign of every term inside: −(4y − 5) = −4y + 5.
展开括号依据分配律。例如 5(2x + 3) = 10x + 15，以及 2x(3x − 4) = 6x² − 8x。若括号外有负号，括号内每一项的符号都要变号：−(4y − 5) = −4y + 5。
a + 3). For 10x² − 5x, factor out 5x to obtain 5x(2x − 1). Always check by expanding back.
因式分解是展开的逆运算。提取各项的最大公因数。在 6a + 9 中，HCF 为 3，得 6a + 9 = 3(2a + 3)。对于 10x² − 5x，提取 5x 得到 5x(2x − 1)。完成后可展开验证。
4. Solving Equations | 解方程
eration on both sides. For x + 5 = 12, subtract 5: x = 7. For 3x = 21, divide by 3: x = 7.
线性方程中变量的指数为 1。我们通过等式两边同时进行相同运算来求解未知数。如 x + 5 = 12，两边减 5 得 x = 7；3x = 21 两边除以 3 得 x = 7。
Two‑ th sides, such as 4m + 2 = 2m + 10, can be solved by collecting m terms on one side: 4m − 2m = 10 − 2 → 2m = 8 → m = 4.
两步方程需要逆推两个运算。解 2y − 3 = 11 时，先加 3 得 2y = 14，再除以 2 得 y = 7。当变量出现在方程两边时，如 4m + 2 = 2m + 10，可将含 m 的项移至一边：4m − 2m = 10 − 2 → 2m = 8 → m = 4。
Always verify the solution by substituting it back into the original equation. This habit catches arithmetic mistakes and deepens understanding of the equality concept.
养成将解代入原方程验算的习惯，既可发现计算错误，又能加深对等量关系的理解。
5. Sequences | 数列
uence, the difference between consecutive terms is constant. For the pattern 4, 7, 10, 13, …, the common difference is +3. The position‑to‑ term, is expressed as 3n + 1, because when n = 1, 3(1) + 1 = 4, n = 2 gives 7, and so on.
数列是按规律排列的一列数。等差数列中相邻两项的差为常数。在 4, 7, 10, 13, … 中，公差为 +3。第 n 项公式为 3n + 1，因为 n = 1 时 3(1) + 1 = 4，n = 2 时得 7，以此类推。
Other sequences may involve square or cube numbers, or patterns that grow by adding consecutive numbers. Recognising these structures allows you to predict future terms and form the nth term rule. For instance, 1, 4, 9, 16, 25, … follows n², while 1, 8, 27, 64, … follows n³.
其他数列可能涉及平方数、立方数，或逐项累加递增的模式。识别这些结构后就能预测后续项并建立第 n 项公式。例如 1, 4, 9, 16, 25, … 遵循 n²，而 1, 8, 27, 64, … 遵循 n³。
The term‑to‑term rule describes how to move from one term to the next, whereas the position‑to‑term rule links the term directly to its position number. Both views are useful for problem‑solving.
逐项规则描述如何从一项得到下一项，而位项规则直接将项与其位置编号联系起来。这两种视角对解决问题都很有用。
6. Ratio and Proportion | 比和比例
simplified just like fractions: 20 : 30 = 2 : 3 (dividing both sides by 10). When sharing £56 in the ratio 3 : 5, the total parts are 8, so one part is £56 ÷ 8 = £7, giving £21 and £35.
比用于比较同类量。3 : 4 表示每 3 单位的第一种量对应 4 单位的第二种量。比可以像分数一样化简：20 : 30 = 2 : 3 (两边同除以 10)。将 £56 按 3 : 5 分配时，总份数为 8，每份 £56 ÷ 8 = £7，因此得到 £21 和 £35。
Proportion describes a multiplicative relationship between two quantities. Two quantities are in direct proportion if they increase or decrease at the same rate. If 5 apples cost £2.50, then 1 apple costs £0.50, and 8 apples cost £4.00. The ratio of apples to cost remains constant.
比例描述两个量之间的倍数关系。若两个量以相同速率增加或减少，则它们成正比例。5 个苹果售价 £2.50，则 1 个苹果 £0.50，8 个苹果 £4.00。苹果数量与成本的比值保持恒定。
The unitary method — finding the value of one unit first — is a powerful strategy for proportion questions. It turns complex comparisons into simple multiplications.
单位法——先求出单一单位的对应量——是解决比例问题的有力工具，能将复杂比较简化为乘法运算。
7. Geometry of Shapes | 图形的几何性质
Angles are measured in degrees (°) and classified by size: acute (0°–90°), right (90°), obtuse (90°–180°) and reflex (180°–360°). The sum of angles on a straight line is 180°, around a point is 360°, and vertically opposite angles are equal. These facts allow you to find missing angles without a protractor.
角以度 (°) 为单位，按大小分为锐角 (0°–90°)、直角 (90°)、钝角 (90°–180°) 和优角 (180°–360°)。平角为 180°，周角为 360°，对顶角相等。运用这些规律可以不解量角器直接求出未知角。
In parallel lines, alternate angles are equal, corresponding angles are equal, and interior (co‑interior) angles sum to 180°. Recognising the Z‑shape for alternate angles and the F‑shape for corresponding angles helps identify them quickly.
平行线中，内错角相等，同位角相等，同旁内角之和为 180°。借助 Z 形识别内错角、F 形识别同位角，可快速判断角度关系。
The interior angles of a triangle always add up to 180°. An equilateral triangle has three 60° angles and equal sides; an isosceles triangle has two equal sides and two equal base angles; a right‑angled triangle contains one 90° angle.
三角形内角和恒为 180°。等边三角形三个角都是 60° 且三边相等；等腰三角形有两边相等，两底角相等；直角三角形含一个 90° 角。
8. Measurement: Perimeter, Area and Volume | 测量：周长、面积和体积
Perimeter is the distance around the outside of a 2‑ a = ½ × base × perpendicular height, parallelogram area = base × perpendicular height, and trapezium area = ½ × (a + b) × height, where a and b are the parallel sides.
周长是二维图形边界的总长。矩形周长 = 2(长 + 宽)。面积度量图形内部的表面大小。矩形面积 = 长 × 宽，三角形面积 = ½ × 底 × 高，平行四边形面积 = 底 × 高，梯形面积 = ½ × (上底 + 下底) × 高。
Compound shapes are made up of several simpler figures. Divide the shape into rectangles, triangles or parts of circles, calculate each area separately and add or subtract as needed. Careful labelling of dimensions prevents errors.
组合图形由多个简单图形拼接而成。可将其分割为矩形、三角形或圆的一部分，分别计算面积后相加或相减。清晰地标注各边长可避免错误。
Volume measures the space inside a 3‑ the total area of all the faces. For a cube of side s, volume = s³ and surface area = 6s².
体积度量三维立体内部的空间。长方体体积 = 长 × 宽 × 高。表面积是所有面的面积总和。棱长为 s 的立方体，体积 = s³，表面积 = 6s²。
9. Statistics | 统计
edian is the middle value when the list is ordered. The mode is the most frequent value, and the range = highest value − lowest value.
数据可通过平均数和离散度来概括。平均数 (均值) 为所有数据之和除以数据个数。中位数是数据排序后中间的值。众数是出现次数最多的值，极差 = 最大值 − 最小值。
otal of 80 corresponds to (20/80) × 360° = 90°. When drawing charts, always label axes and provide a title.
条形图、象形图和饼图可直观展示数据。饼图以角度表示比例：若总数为 80，某类频数为 20，则扇形角 = (20/80) × 360° = 90°。绘制图表时务必标注坐标轴和标题。
Comparing distributions often involves looking at both the average and the range. Two classes might have the same mean test score, but the one with the smaller range is more consistent. Understanding this distinction helps in interpreting real‑world data.
比较分布时常需同时考虑平均值与极差。两个班级可能有相同的平均考试分数，但极差更小的班级成绩更稳定。理解这一区别有助于解读现实数据。
10. Introduction to Probability | 概率入门
s a fraction, decimal or percentage between 0 (impossible) and 1 (certain). For a fair six‑sided die, P(rolling a 4) = 1/6. The sum of probabilities of all possible outcomes is 1.
概率衡量事件发生的可能性，用 0 (不可能) 到 1 (必然) 之间的分数、小数或百分比表示。对一颗均匀的六面骰子，P(掷出 4) = 1/6。所有可能结果的概率之和为 1。
If P(heads) = ½ and a coin is flipped 200 times, we expect about 100 heads. This is a long‑run average, not a guarantee for a specific experiment.
事件发生的期望次数等于概率乘以试验次数。若 P(正面) = ½，抛硬币 200 次，则期望约出现 100 次正面。这是长期平均概念，并非某次特定实验的保证。
sums and identify probabilities: P(total = 7) = 6/36 = 1/6 because six pairs (1,6), (2,5), … (6,1) give a sum of 7.
样本空间图表系统列出所有可能的结果。掷两个骰子共有 36 种等可能组合。绘制网格有助于直观呈现点数和并计算概率：P(和为 7) = 6/36 = 1/6，因为有 6 组如 (1,6)、(2,5) … (6,1) 的结果。
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