📚 Year 11 OCR Maths: Core Topics Revision | Year 11 OCR 数学：核心知识点梳理
This revision guide covers the essential topics in the OCR GCSE Mathematics syllabus for Year 11 students. Whether you are sitting the Foundation or Higher tier, a solid understanding of these core areas—Number, Algebra, Geometry, Statistics, and more—will boost your confidence and exam performance. Let's break down each topic with clear bilingual explanations and key formulae.
本复习指南涵盖 Year 11 OCR GCSE 数学大纲中的核心知识点。无论你参加基础还是高级考卷，扎实掌握数、代数、几何、统计等关键领域将提升你的信心与考试成绩。让我们用清晰的双语解释和关键公式逐一梳理。
1. Number: Fractions, Decimals and Percentages | 数：分数、小数与百分数
convert freely between fractions, decimals and percentages. For instance, 3/4 = 0.75 = 75%.
在 OCR 考试中，你必须能在分数、小数和百分数之间自由转换。例如 3/4 = 0.75 = 75%。
o add or subtract fractions, find a common denominator. E.g., 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
加减分数时，要找到公分母。例如 1/2 + 1/3 = 3/6 + 2/6 = 5/6。
A percentage increase or decrease is calculated using: Percentage change = (change ÷ original) × 100 2, the increase is £12, so the percentage increase is (12 ÷ 40) × 100 = 30%.
百分比增加或减少的计算公式为：百分比变化 = (变化量 ÷ 原始量) × 100。例如价格从 40 英镑涨到 52 英镑，增加 12 英镑，百分比增加为 (12 ÷ 40) × 100 = 30%。
Repeated percentage change uses a multiplier. For a 5% increase, the multiplier is 1.05. For a 20% use 0.8. Compound interest follows the formula: Amount = Principal × (1 + rate/100)n, where n is the number of years.
重复百分比变化使用乘数。5% 增加乘数为 1.05；20% 减少使用 0.8。复利公式为：金额 = 本金 × (1 + 利率/100)n，其中 n 是年数。
2. Number: Standard Form and Estimation | 数：标准形式与估算
Standard form is written as A × 10n A < 10 and n is an integer. E.g., 4500 = 4.5 × 10³, 0.0062 = 6.2 × 10−3.
标准形式写作 A × 10n，其中 1 ≤ A < 10，n 为整数。例如 4500 = 4.5 × 10³，0.0062 = 6.2 × 10−3。
To multiply numbers in standard form, multiply the A parts and add the powers of 10. Division subtracts the powers.
标准形式乘法：A 部分相乘，10 的幂指数相加。除法减去指数。
Significant figures are used to approximate. Round 0.004567 to 2 s.f. gives 0.0046. Estimating before calculating helps check answers.
有效数字用于近似。将 0.004567 四舍五入到 2 位有效数字得 0.0046。计算前估算有助于检查答案。
Standard form: a × 10b, 1 ≤ a < 10
3. Algebra: Expressions, Equations and Inequalities | 代数：表达式、方程与不等式
fying expressions involves collecting like terms. For example, 3x + 2y - x + 5y = 2x + 7y.
化简表达式包括合并同类项。例如 3x + 2y - x + 5y = 2x + 7y。
Expanding brackets: a(b + c) = ab + ac. Also, (x + a)(x + b) = x² + (a+b)x + ab.
展开括号：a(b + c) = ab + ac。还有 (x + a)(x + b) = x² + (a+b)x + ab。
lving linear equations: isolate the unknown. E.g., 2x - 5 = 9 → 2x = 14 → x = 7.
解线性方程：分离未知数。例如 2x - 5 = 9 → 2x = 14 → x = 7。
x + 6 = 0 → (x - 2)(x - 3) = 0 → x = 2 or x = 3. Or using the quadratic formula:
二次方程（高级）可通过因式分解求解：x² - 5x + 6 = 0 → (x - 2)(x - 3) = 0 → x = 2 或 x = 3。或使用求根公式：
x = [−b ± √(b² − 4ac)] / 2a
Inequalities are solved like equations but remember to flip the sign when multiplying or dividing by a negative number. E.g., -2x > 8 → x < -4.
不等式解法类似方程，但当乘以或除以负数时要翻转不等号。例如 -2x > 8 → x < -4。
4. Algebra: Graphs and Functions | 代数：图像与函数
s the y-intercept. For example, y = 2x + 3 crosses the y-axis at (0,3) and has gradient 2.
直线图像方程为 y = mx + c，其中 m 是斜率，c 是 y 截距。例如 y = 2x + 3 在 (0,3) 与 y 轴相交，斜率为 2。
Quadratic graphs are U-shaped (parabolas). The turning point is either a minimum or maximum. The roots are where the graph crosses the x-axis.
二次函数图像是 U 形的（抛物线）。转折点是最小值或最大值。根是图像与 x 轴的交点。
Simultaneous equations can be solved graphically by finding the point of intersection. Algebraically, by elimination or substitution.
联立方程可以通过寻找图像的交点来求解。也可用代数法：消元法或代入法。
For Higher tier, understand cubic and reciprocal graphs, and the concept of a function f(x). Inverse functions and composite functions may also appear.
高级卷还需了解三次和倒数图像，以及函数 f(x) 的概念。反函数和复合函数也可能出现。
5. Ratio, Proportion and Rates of Change | 比率、比例与变化率
Ratios compare quantities. Simplify ratios by dividing by common factors. E.g., 10:15 simplifies to 2:3.
比率用于比较数量。通过除以公因数来化简比率。例如 10:15 可化简为 2:3。
arts = sum of ratio parts. For a ratio 3:5, total parts = 8, so one share = total ÷ 8.
按给定比例分配：总份数等于比率各项之和。若比率为 3:5，总份数为 8，因此一份等于总数 ÷ 8。
oportion means y = kx; if x doubles, y doubles. Inverse proportion (Higher) means y = k/x.
正比例指 y = kx；若 x 翻倍，y 也翻倍。反比例（高级）指 y = k/x。
sity = mass ÷ volume, pressure = force ÷ area. Use formula triangles to rearrange.
速度 = 距离 ÷ 时间，密度 = 质量 ÷ 体积，压强 = 力 ÷ 面积。可利用公式三角形进行变形。
6. Geometry: Angles and Polygons | 几何：角度与多边形
Angles on a straight line sum to 180°; angles around a point sum to 360°. Vertically opposite angles are equal.
直线上的角之和为 180°；绕一点旋转的角之和为 360°。对顶角相等。
In polygons, interior angle sum = (n - 2) × 180°, where n is
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