Essential Maths 9H Key Concepts Review | Essential Maths 9H 知识点精讲

📚 Essential Maths 9H Key Concepts Review | Essential Maths 9H 知识点精讲

Essential Maths 9H is a core textbook for Key Stage 3 Year 9 students aiming at Higher tier GCSE preparation. This article distils its most important topics – from indices and algebra to Pythagoras and trigonometry – into clear, bilingual explanations. Each section pairs an English explanation with a matching Chinese one, helping you master the material quickly and effectively.

《Essential Maths 9H》是面向英国 KS3 九年级高阶学生的核心教材，旨在衔接 GCSE 高阶内容。本文从指数运算、代数推理到毕达哥拉斯定理和三角函数，提炼出最核心的知识点，采用中英对照方式讲解，帮你高效攻克每一个主题。

1. Integers, powers and roots | 整数、幂与方根

In Year 9 Higher, you must be confident with negative numbers, BIDMAS, and the laws of indices for positive and negative exponents. A power like 5³ means 5 × 5 × 5 = 125. A negative index means a reciprocal: 2⁻³ = 1/2³ = 1/8. The square root symbol √ always gives the principal (positive) root, e.g. √49 = 7, but x² = 49 has solutions x = ±7.

在九年级高阶，你必须熟练掌握负数运算、运算顺序以及正负指数的运算法则。5³ 表示三个 5 相乘得 125。负指数是倒数关系：2⁻³ = 1/2³ = 1/8。平方根符号 √ 只表示算术平方根，如 √49 = 7，但方程 x² = 49 有两个解 x = ±7。

The three core index laws are: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ. You can also raise a product to a power: (ab)ⁿ = aⁿbⁿ. These laws are essential when simplifying algebraic expressions or working with standard form.

三条核心指数定律是：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。此外还可以处理积的乘方：(ab)ⁿ = aⁿbⁿ。这些定律在化简代数式和标准形式运算中不可或缺。

2⁴ × 2³ = 2⁷ = 128, (3²)³ = 3⁶ = 729, a⁻¹ = 1/a

2. Standard form and surds | 标准形式与根式

Standard form writes numbers as a × 10ⁿ where 1 ≤ a < 10. For example, 0.00057 = 5.7 × 10⁻⁴. You need to add, subtract, multiply and divide numbers in standard form without a calculator, paying close attention to the powers of ten.

标准形式将数字写作 a × 10ⁿ，其中 1 ≤ a < 10。例如 0.00057 = 5.7 × 10⁻⁴。你需要能够在没有计算器的情况下对标准形式的数进行加减乘除，并特别留意 10 的指数。

Surds are irrational roots like √2 or √3, left in root form to keep exactness. You simplify surds by extracting square factors: √50 = √(25 × 2) = 5√2. Always rationalise the denominator where required: 1/√2 = √2/2.

根式是指像 √2 或 √3 这样的无理数，保留根号以保证精确值。化简时提取平方因子：√50 = √(25 × 2) = 5√2。同时需要将分母有理化：1/√2 = √2/2。

3. Algebraic expressions and brackets | 代数式与括号展开

You must be able to expand and simplify linear, and later quadratic, expressions. For single brackets: 5(3x − 2) = 15x − 10. For double brackets: (2x + 3)(x − 4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12. Recognise patterns such as (a + b)² = a² + 2ab + b².

你需熟练掌握线性及二次代数式的展开与化简。单项乘括号：5(3x − 2) = 15x − 10。展开双重括号：(2x + 3)(x − 4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12。熟记完全平方公式：(a + b)² = a² + 2ab + b²。

Factorising is the reverse process. For 6x² + 9x, the common factor is 3x → 3x(2x + 3). For quadratics, find two numbers that multiply to give the constant term and add to the coefficient of x.

因式分解是展开的逆运算。如 6x² + 9x，公因式为 3x，分解得 3x(2x + 3)。对于二次三项式，找出两个数使得乘积为常数项，和为 x 的系数。

4. Linear equations, formulae and inequalities | 线性方程、公式变形与不等式

To solve an equation like 4x − 7 = 2x + 9, collect like terms: 4x − 2x = 9 + 7 → 2x = 16 → x = 8. Always perform the same operation on both sides. Solve equations with fractions by multiplying through by the common denominator.

解方程如 4x − 7 = 2x + 9，需移项合并：4x − 2x = 9 + 7 → 2x = 16 → x = 8。牢记方程两边同做相同运算。含分数的方程可先乘以公分母消去分母。

Rearranging formulae involves making a different variable the subject. For v = u + at, making t the subject gives t = (v − u)/a. Inequalities work similarly to equations, but reverse the sign when multiplying or dividing by a negative number: −2x > 6 → x < −3.

公式变形是指将一个变量表示为主题。例如 v = u + at，将 t 作为主题得 t = (v − u)/a。不等式的解法和方程相似，但当两边乘或除以一个负数时，不等号方向要改变：−2x > 6 → x < −3。

5. Sequences and the nth term | 数列与通项公式

Generate linear sequences from an nth term rule such as 3n + 2: for n = 1,2,3… we get 5, 8, 11… The difference between consecutive terms is constant and equals the coefficient of n. Reverse the process by finding a rule from given terms: if the sequence is 7, 11, 15, 19…, the term-to-term difference is 4, so nth term = 4n + 3.

根据通项公式生成线性数列，例如 3n + 2：当 n = 1,2,3…，数列为 5, 8, 11… 相邻项的差为常数，且等于 n 的系数。反过来，由数列找出规律：数列 7, 11, 15, 19… 的差为 4，故通项为 4n + 3。

Quadratic sequences have a second difference that is constant. For the square numbers 1, 4, 9, 16…, the general term is n². Harder sequences like 3, 6, 11, 18… have nth term n² + 2.

二次数列的二级差为常数。平方数数列 1, 4, 9, 16… 的通项为 n²。较复杂的如 3, 6, 11, 18… 通项是 n² + 2。

Linear: common first difference → nth term of form an + b
Quadratic: common second difference → nth term of form an² + bn + c
线性数列：一级差恒定 → 通项形式 an + b
二次数列：二级差恒定 → 通项形式 an² + bn + c

6. Ratio, proportion and rates | 比、比例与速率

Ratios compare parts to parts, and can be simplified by dividing by a common factor. A recipe ratio of 3:2:1 for flour, sugar and butter means 300 g : 200 g : 100 g for 600 g total. Divide a quantity in a given ratio by finding the value of one share first: split £120 in ratio 3:5 → 3 + 5 = 8 shares, one share = £15, so £45 : £75.

比用于比较各部分的大小，可除以公因数进行化简。食谱中面粉、糖和黄油的比是 3:2:1，总重 600 g 对应 300 g : 200 g : 100 g。按给定比例分配数量时，先求一份的量：把 £120 按 3:5 分 → 总份数 8，一份 £15，得 £45 : £75。

Direct proportion means y = kx, where k is the constant of proportionality. Graphically it is a straight line through the origin. Inverse proportion y = k/x gives a hyperbola. In rates, speed = distance ÷ time, and density = mass ÷ volume.

正比例关系 y = kx，k 为比例常数，图像为过原点的直线。反比例 y = k/x 图像为双曲线。关于速率：速度 = 距离 ÷ 时间，密度 = 质量 ÷ 体积。

7. Percentages and compound change | 百分数与复合变化

Find a percentage of an amount, increase or decrease by a percentage, and use multipliers. A 15% increase means multiply by 1.15; a 20% decrease means multiply by 0.8. Reverse percentages: if a price after a 10% discount is £36, the original was £36 ÷ 0.9 = £40.

会求一个数量的百分之几，进行百分数增减，并使用乘数。增加 15% 即乘以 1.15；减少 20% 即乘以 0.8。逆向百分数：如果打九折后价格 £36，原价为 £36 ÷ 0.9 = £40。

Compound interest formula: A = P(1 + r/100)ⁿ, where A is amount, P is principal, r is rate, n is number of years. Depreciation works similarly but uses a multiplier less than 1, e.g. 0.85 for a 15% loss per year.

复利公式：A = P(1 + r/100)ⁿ，A 为本利和，P 为本金，r 为年利率，n 为年数。贬值同理，但乘数小于 1，如每年损失 15%，乘数为 0.85。

£500 invested at 4% for 3 years → 500 × (1.04)³ ≈ £562.43

8. Angles, lines and polygons | 角、线与多边形

Angles on a straight line add to 180°, around a point to 360°. Vertically opposite angles are equal. Alternate and corresponding angles on parallel lines are equal. Interior angles (co-interior) sum to 180°.

直线上的角之和为 180°，绕一点的周角为 360°。对顶角相等。平行线中的内错角、同位角均相等，同旁内角之和为 180°。

For any polygon, the sum of exterior angles is always 360°. Interior angle sum = (n − 2) × 180° for an n-sided polygon. A regular polygon has equal sides and angles, so each interior angle = (n − 2) × 180° / n.

任意多边形的外角和恒为 360°。内角和 = (n − 2) × 180°。正多边形各边各角均等，每个内角为 (n − 2) × 180° / n。

Polygon	Sides	Interior angle sum
Triangle	3	180°
Quadrilateral	4	360°
Pentagon	5	540°
Hexagon	6	720°

9. Pythagoras’ Theorem | 毕达哥拉斯定理

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². The hypotenuse is always opposite the right angle. To find a shorter side, rearrange: a² = c² − b².

在直角三角形中，斜边的平方等于两条直角边的平方和：a² + b² = c²。斜边始终对着直角。求直角边时调整公式：a² = c² − b²。

Apply the theorem to calculate distances between two coordinates (x₁, y₁) and (x₂, y₂): length = √[(x₂ − x₁)² + (y₂ − y₁)²]. This is the distance formula derived from a right triangle on the grid.

应用该定理可计算坐标平面上两点 (x₁, y₁) 和 (x₂, y₂) 的距离：长度 = √[(x₂ − x₁)² + (y₂ − y₁)²]。这是从网格上的直角三角形推导出的距离公式。

Example: sides 6 cm and 8 cm → hypotenuse = √(6² + 8²) = √100 = 10 cm

10. Trigonometry: right-angled triangles | 三角学：直角三角形

The three trigonometric ratios relate sides and acute angles: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Remember SOH CAH TOA. Always label the sides relative to the given angle first.

三个三角函数比将边长与锐角联系起来：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。牢记 SOH CAH TOA。首先要根据给定角标出各边。

Use the inverse functions sin⁻¹, cos⁻¹ and tan⁻¹ on a calculator to find an angle. When finding a missing side, choose the ratio that links the known length and the unknown side. For example, if angle = 35° and adjacent = 12 cm, then hypotenuse = 12 / cos 35°.

使用计算器上的反函数 sin⁻¹、cos⁻¹ 和 tan⁻¹ 求角度。求未知边长时，选择能连接已知边与未知边的比。例如，已知角 35° 和邻边 12 cm，则斜边 = 12 / cos 35°。

sin 30° = ½, cos 60° = ½, tan 45° = 1
Exact values for 0°, 30°, 45°, 60° and 90° should be memorised.
sin 30° = ½, cos 60° = ½, tan 45° = 1
熟记 0°、30°、45°、60°、90° 的精确三角函数值。

11. Perimeter, area and volume | 周长、面积与体积

Area of triangle = ½ × base × height. Area of trapezium = ½ (a + b) × height, where a and b are the parallel sides. Circumference of circle = πd or 2πr, area = πr². Use 3.14 or the π key on a calculator.

三角形面积 = ½ × 底 × 高。梯形面积 = ½ (a + b) × 高，其中 a、b 为上底和下底。圆的周长 = πd 或 2πr，面积 = πr²。计算时可用 3.14 或计算器上的 π 键。

Volume of a prism = area of cross-section × length. For a cylinder, volume = πr²h. Surface area is the total area of all faces. For compound shapes, split into simpler parts, calculate separately, then sum.

棱柱的体积 = 横截面积 × 长。圆柱体积 = πr²h。表面积为所有面的面积之和。对于组合图形，可分割成基本图形分别计算再求和。

Cone volume = ⅓πr²h, sphere volume = ⁴⁄₃πr³, surface area of sphere = 4πr²

12. Probability and statistics | 概率与统计

Probability of an event = number of favourable outcomes / total possible outcomes. It is always a number between 0 and 1. Probabilities of all possible outcomes sum to 1. Expected frequency = probability × number of trials.

事件概率 = 有利结果数 / 所有可能结果数。概率始终在 0 到 1 之间。所有可能结果的概率之和为 1。期望频数 = 概率 × 试验次数。

Sample space diagrams list all outcomes for two events, such as spinning two spinners. Frequency trees and two-way tables help organise data. The mean, median, mode and range summarise data. Outliers can affect the mean significantly.

样本空间图列出两个事件的所有可能结果，如两次转盘。频率树和双向表有助于整理数据。用平均数、中位数、众数和极差概括数据。异常值对平均值影响较大。

Scatter graphs show correlation between two variables. Draw a line of best fit to make predictions. Positive correlation means both increase together; negative correlation means one increases as the other decreases.

散点图显示两个变量间的相关关系。画最佳拟合线可进行预测。正相关意味着两者同时增大；负相关则是一个增大时另一个减小。

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Essential Maths 9H Key Concepts Review | Essential Maths 9H 知识点精讲