📚 Year 9 OCR Mathematics: Core Knowledge Review | 9年级OCR数学：核心知识点梳理
This article provides a structured revision of the essential topics in Year 9 OCR Mathematics. It covers number skills, algebra, geometry, ratio, probability, and statistics, helping students consolidate fundamental concepts before moving on to GCSE preparation. Each section is presented in both English and Chinese to support bilingual learners.
本文系统梳理了Year 9 OCR数学的核心知识点，涵盖数字运算、代数、几何、比率、概率和统计等内容，帮助学生在进入GCSE阶段前打好坚实基础。每节均提供中英双语讲解，便于双语学习者对照理解。
1. Number: Fractions, Decimals and Percentages | 数字：分数、小数与百分比
Fluency in converting between fractions, decimals and percentages is vital. For example, 1/2 = 0.5 = 50%. To convert a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a percentage, multiply by 100.
分数、小数和百分比之间的熟练转换非常重要。例如 1/2 = 0.5 = 50%。将分数化为小数时，用分子除以分母；将小数化为百分数时，乘以100。
When adding or subtracting fractions, find a common denominator first. For 1/3 + 1/4, the common denominator is 12, giving 4/12 + 3/12 = 7/12.
分数的加减法需要先找到公分母。例如 1/3 + 1/4，公分母是12，得到 4/12 + 3/12 = 7/12。
Multiplying fractions is straightforward: multiply numerators and multiply denominators. 2/5 × 3/7 = 6/35. Dividing by a fraction is equivalent to multiplying by its reciprocal: 2/5 ÷ 3/7 = 2/5 × 7/3 = 14/15.
分数乘法直接将分子相乘、分母相乘：2/5 × 3/7 = 6/35。分数除法等于乘以倒数：2/5 ÷ 3/7 = 2/5 × 7/3 = 14/15。
Ordering fractions, decimals and percentages requires changing them into the same form. Compare 3/5, 0.59 and 58% by converting all to decimals: 0.6, 0.59, 0.58. Hence 3/5 is largest.
比较分数、小数和百分比时，需统一形式。比较 3/5, 0.59 和 58%，全部化为小数：0.6, 0.59, 0.58，因此 3/5 最大。
2. Powers and Roots | 幂与根
Squares, cubes and higher powers follow index laws. 5² = 25, 4³ = 64. The square root of 49 is 7, and the cube root of 27 is 3.
平方、立方和高次幂遵循指数运算法则。5² = 25，4³ = 64。49的平方根是7，27的立方根是3。
Multiplying powers with the same base: aᵐ × aⁿ = aᵐ⁺ⁿ. For example, 2³ × 2⁴ = 2⁷ = 128. Dividing: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. (5⁶) ÷ (5²) = 5⁴.
同底数幂相乘：aᵐ × aⁿ = aᵐ⁺ⁿ。例如 2³ × 2⁴ = 2⁷ = 128。相除：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。(5⁶) ÷ (5²) = 5⁴。
A power raised to another power: (aᵐ)ⁿ = aᵐⁿ. (3²)³ = 3⁶ = 729. Any number to the power of zero is 1: 8⁰ = 1.
幂的乘方：(aᵐ)ⁿ = aᵐⁿ。(3²)³ = 3⁶ = 729。任何非零数的零次幂等于1：8⁰ = 1。
Negative indices produce reciprocals: a⁻ⁿ = 1/aⁿ. 2⁻³ = 1/2³ = 1/8. Fractional indices relate to roots: a^(½) = √a, a^(⅓) = ∛a. So 16^(½) = 4.
负指数表示倒数：a⁻ⁿ = 1/aⁿ。2⁻³ = 1/2³ = 1/8。分数指数与根式相关：a^(½) = √a，a^(⅓) = ∛a。因此 16^(½) = 4。
Standard form expresses very large or small numbers as a × 10ⁿ, where 1 ≤ a < 10. Example: 4,500 = 4.5 × 10³, and 0.0006 = 6 × 10⁻⁴.
科学记数法将极大或极小的数表示为 a × 10ⁿ，其中 1 ≤ a < 10。例如：4,500 = 4.5 × 10³，0.0006 = 6 × 10⁻⁴。
3. Algebraic Expressions and Simplification | 代数表达式与化简
Collect like terms by adding or subtracting coefficients: 3x + 2y – x + 5y = 2x + 7y. Terms with different variables or powers cannot be combined.
合并同类项时将系数相加减：3x + 2y – x + 5y = 2x + 7y。不同变量或不同幂次的项不能合并。
Expand brackets by multiplying each term inside the bracket. 4(x + 3) = 4x + 12. Double brackets: (x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10.
去括号时将括号外的因数乘入括号内每一项。4(x + 3) = 4x + 12。双括号展开：(x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10。
Factorise by extracting the highest common factor (HCF). 6x + 9 = 3(2x + 3). For quadratics like x² + 5x + 6, find two numbers that multiply to 6 and add to 5: (x + 2)(x + 3).
因式分解时要提取最大公因数。6x + 9 = 3(2x + 3)。对于二次式如 x² + 5x + 6，找到乘积为6、和为5的两数：(x + 2)(x + 3)。
Use index laws in algebra: x² × x³ = x⁵. Divide terms by subtracting indices: 8x⁵ ÷ 2x² = 4x³. Remember that x = x¹.
在代数中应用指数法则：x² × x³ = x⁵。相除时指数相减：8x⁵ ÷ 2x² = 4x³。记住 x = x¹。
4. Solving Linear Equations and Inequalities | 解线性方程与不等式
Solve equations by performing inverse operations to isolate the variable. For 3x + 4 = 19: subtract 4 to get 3x = 15, then divide by 3: x = 5.
通过逆运算解方程，将变量孤立出来。解 3x + 4 = 19：两边减4得 3x = 15，再除以3得 x = 5。
When variables appear on both sides, collect them on one side. 5x – 3 = 2x + 9 → 5x – 2x = 9 + 3 → 3x = 12 → x = 4.
变量在等号两边时，将其移到同一边。5x – 3 = 2x + 9 → 5x – 2x = 9 + 3 → 3x = 12 → x = 4。
Inequalities are solved similarly but remember to reverse the sign when multiplying or dividing by a negative number. -2x ≤ 8 gives x ≥ -4.
不等式的解法类似，但乘以或除以负数时要反向改变不等号。 -2x ≤ 8 变为 x ≥ -4。
Represent inequalities on a number line: use an open circle for < or >, and a closed circle for ≤ or ≥. Graph compound inequalities such as -2 < x ≤ 3 with appropriate markings.
在数轴上表示不等式：< 或 > 用空心圆，≤ 或 ≥ 用实心圆。画出复合不等式如 -2 < x ≤ 3，要正确标记端点。
5. Sequences and the nth Term | 数列与第n项
An arithmetic sequence has a constant difference, d, between consecutive terms. For 2, 6, 10, 14, …, d = 4. The nth term formula is aₙ = a₁ + (n-1)d.
等差数列相邻两项之间的差d为常数。数列 2, 6, 10, 14, …，d = 4。第n项公式为 aₙ = a₁ + (n-1)d。
To find the nth term from a sequence, identify the first term and the common difference. Sequence 7, 13, 19, 25: first term 7, difference 6, so nth term = 7 + (n-1)6 = 6n + 1.
已知数列求第n项时，先找出首项和公差。数列 7, 13, 19, 25：首项7，公差6，第n项 = 7 + (n-1)6 = 6n + 1。
Use the nth term to generate any term. For nth term 3n – 4, the 10th term is 3(10) – 4 = 26. Check if a number is in the sequence: solve 3n – 4 = 35 → 3n = 39 → n = 13, so 35 is the 13th term.
利用第n项公式可生成任意项。若第n项为 3n – 4，第10项为 3×10 – 4 = 26。判断某数是否在数列中：解 3n – 4 = 35 → 3n = 39 → n = 13，因此35是第13项。
Other sequences include geometric progressions (multiplying by a constant ratio) and special sequences like square numbers (1, 4, 9, 16…) or triangular numbers.
其他类型的数列包括等比数列（乘以固定比值），以及特殊数列如平方数（1, 4, 9, 16…）和三角形数。
6. Graphs and Coordinates | 图表与坐标
Plot points on a Cartesian plane using (x, y) coordinates. The x-coordinate moves horizontally, the y-coordinate vertically. The point (3, -2) is 3 units right and 2 units down from the origin.
在直角坐标系中用 (x, y) 描点。x坐标表示水平移动，y坐标表示垂直移动。点 (3, -2) 表示从原点向右3、向下2。
Linear graphs represent equations of the form y = mx + c, where m is the gradient and c is the y-intercept. For y = 2x + 1, the line crosses the y-axis at 1 and has slope 2.
线性图像表示形如 y = mx + c 的方程，其中 m 是斜率，c 是y轴截距。y = 2x + 1 的图像过点 (0,1)，斜率为2。
Gradient is calculated as rise over run: m = (change in y) / (change in x). Between (1,3) and (4,9), m = (9-3)/(4-1) = 6/3 = 2.
斜率可用垂直变化除以水平变化计算：m = (y的变化量)/(x的变化量)。点 (1,3) 和 (4,9) 之间，m = (9-3)/(4-1) = 6/3 = 2。
To find the equation of a line, use y = mx + c and substitute a known point to solve for c. A line with gradient 3 passing through (2,5): y = 3x + c → 5 = 3(2) + c → c = -1, so equation is y = 3x – 1.
求直线方程时，使用 y = mx + c 形式，代入已知点求c。斜率为3且过点(2,5)的直线：y = 3x + c → 5 = 3×2 + c → c = -1，方程为 y = 3x – 1。
7. Ratio and Proportion | 比率与比例
Ratio compares quantities. Simplify ratios by dividing by the highest common factor. 18:24 simplifies to 3:4 (divide both by 6). Keep units consistent when forming ratios.
比率用于比较数量。化简比率时除以最大公因数。18:24 化简为 3:4（同除以6）。列出比率时要保持单位一致。
Sharing in a ratio means dividing a quantity into parts. Share £60 in the ratio 2:3: total parts = 5, one part = £60 ÷ 5 = £12, so amounts are 2×12 = £24 and 3×12 = £36.
按比例分配是将总量按份数拆分。按 2:3 分 £60：总份数 = 5，一份 = £60 ÷ 5 = £12，因此分别得 2×12 = £24 和 3×12 = £36。
Direct proportion: as one quantity increases, the other increases at the same rate. y = kx, where k is the constant of proportionality. If 4 notebooks cost £7, 10 notebooks cost £17.50 (k = 1.75 per notebook).
正比例关系：一个量增加时，另一个量按相同比例增加。y = kx，k为比例常数。若4本笔记本 £7，则10本价格 £17.50（k = 每本1.75）。
Use the unitary method to solve proportion problems: find the value of one unit first, then scale up to the required amount.
解决比例问题可使用归一法：先求出一个单位的值，再扩大到所需数量。
8. Percentages and Percentage Change | 百分比与百分比变化
Find a percentage of an amount without a calculator: 10% of 80 = 8, so 30% = 3 × 8 = 24. With a calculator, multiply by the decimal equivalent: 17% of 250 = 0.17 × 250 = 42.5.
求一个数的百分比心算：80的10%是8，因此30%为 3×8 = 24。用计算器则乘以对应小数：250的17% = 0.17 × 250 = 42.5。
Percentage increase: new value = original × (1 + percentage/100). A £50 item with a 20% increase costs £50 × 1.2 = £60. Decrease: £80 with 15% off is £80 × 0.85 = £68.
百分比增加：新值 = 原值 × (1 + 百分比/100)。£50 的商品涨价20%后为 £50 × 1.2 = £60。减少：£80 打85折为 £80 × 0.85 = £68。
Calculate percentage change: (change ÷ original) × 100%. From 40 to 55: change = 15, percentage increase = (15/40) × 100 = 37.5%.
计算百分比变化：(变化量 ÷ 原值) × 100%。从40到55：变化量15，涨幅 = (15/40) × 100 = 37.5%。
Simple interest: I = P × r × t. £500 at 4% per year for 3 years gives £500 × 0.04 × 3 = £60 interest. Compound interest: A = P(1 + r/100)ⁿ. For the same principal and rate over 3 years: A = 500 × (1.04)³ ≈ £562.43.
单利：利息 = 本金 × 利率 × 时间。£500 年利率4%存3年，利息 = 500 × 0.04 × 3 = £60。复利：终值 = 本金 × (1 + r/100)ⁿ。同等条件3年：终值 = 500 × (1.04)³ ≈ £562.43。
9. Angles in Polygons and Parallel Lines | 多边形内角与平行线
Angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal.
平角为180°，周角为360°，对顶角相等。
When a transversal crosses parallel lines: corresponding angles are equal, alternate angles are equal, and co-interior (allied) angles sum to 180°.
一条截线与平行线相交时：同位角相等，内错角相等，同旁内角互补（和为180°）。
In a triangle, the three interior angles always sum to 180°. An exterior angle equals the sum of the two opposite interior angles.
三角形三内角和恒为180°。一个外角等于两个不相邻内角之和。
For any polygon, the sum of interior angles = (n – 2) × 180°, where n is the number of sides. A pentagon has (5-2)×180° = 540°. The sum of exterior angles of any convex polygon is 360°.
任意多边形的内角和 = (n – 2) × 180°，n为边数。五边形内角和为 (5-2)×180° = 540°。任何凸多边形的外角和均为360°。
10. Pythagoras’ Theorem | 毕达哥拉斯定理
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: a² + b² = c², where c is the longest side.
在直角三角形中，斜边的平方等于两直角边的平方和：a² + b² = c²，其中c为斜边。
To find the hypotenuse: given legs 6 cm and 8 cm, c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
求斜边：已知直角边6 cm和8 cm，c = √(6² + 8²) = √(36+64) = √100 = 10 cm。
To find a shorter side: if c = 13 and a = 5, then b = √(13² – 5²) = √(169 – 25) = √144 = 12.
求直角边：若斜边为13，一直角边为5，则另一直角边 b = √(13² – 5²) = √(169 – 25) = √144 = 12。
Use Pythagoras to solve problems involving distance, such as the diagonal of a rectangle with sides 3 m and 4 m: diagonal = √(3² + 4²) = 5 m.
运用勾股定理解决距离问题，如边长为 3 m 和 4 m 的长方形对角线长度：√(3² + 4²) = 5 m。
11. Area, Perimeter and Volume | 面积、周长与体积
Perimeter of a rectangle: P = 2(l + w). Circumference of a circle: C = 2πr or C = πd. A semi-circle’s perimeter includes the diameter: πr + d.
长方形周长：P = 2(长+宽)。圆的周长：C = 2πr 或 C = πd。半圆的周长包含直径：πr + d。
Area of triangle: ½ × base × height. Area of parallelogram: base × height. Trapezium area: ½ (a + b) h, where a and b are parallel sides.
三角形面积：½ × 底 × 高。平行四边形面积：底 × 高。梯形面积：½ (a + b) h，a和b为平行边。
Circle area: A = πr². For a circle with radius 5 cm, area = π × 25 ≈ 78.5 cm² (using π ≈ 3.14).
圆面积：A = πr²。半径为 5 cm 的圆，面积 = π × 25 ≈ 78.5 cm²（π取3.14）。
Volume of a cuboid: length × width × height. Volume of a prism: area of cross-section × length. Cylinder volume: V = πr²h.
长方体体积：长 × 宽 × 高。棱柱体积：底面积 × 高。圆柱体积：V = πr²h。
Surface area is the total area of all faces. For a cube of side 3 cm, surface area = 6 × (3×3) = 54 cm². For a cylinder, total surface area = 2πr² + 2πrh.
表面积为所有面的面积之和。边长为 3 cm 的正方体表面积 = 6 × (3×3) = 54 cm²。圆柱总表面积 = 2πr² + 2πrh。
12. Probability and Statistics | 概率与统计
Probability measures how likely an event is, ranging from 0 (impossible) to 1 (certain). Probability = number of favourable outcomes / total number of outcomes. Tossing a fair coin: P(heads) = ½.
概率衡量事件发生的可能性，范围从0（不可能）到1（必然）。概率 = 有利结果数 / 所有可能结果数。抛一枚均匀硬币：P(正面) = ½。
For combined events, use sample space diagrams. Two dice: probability of sum 7 is 6/36 = 1/6. Tree diagrams help with sequential events, multiplying along branches.
组合事件可使用样本空间图。两枚骰子点数之和为7的概率是 6/36 = 1/6。树形图用于连续事件，沿分支相乘。
Mean, median, mode and range summarise data sets. Mean = sum of data ÷ number of items. Median is the middle value when ordered. Mode is the most frequent item. Range = maximum – minimum.
平均数、中位数、众数和极差可概括数据集。平均数 = 数据总和 ÷ 数据个数。中位数是排序后居中的值。众数是出现最频繁的值。极差 = 最大值 − 最小值。
Represent data with bar charts, pie charts, and line graphs. Frequency tables and grouped frequency diagrams are used for larger data sets. Interpret charts by reading scales and calculating frequencies.
用条形图、饼图和折线图表示数据。较大数据集可使用频数表和分组频数图。解读图表时需读取刻度并计算频数。
Scatter graphs show correlation between two variables. Line of best fit can be drawn to estimate values. Positive correlation means as one variable increases, the other tends to increase.
散点图显示两个变量之间的相关性。可画出最佳拟合线进行估算。正相关意味着一个变量增大时，另一个也倾向于增大。
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