Year 8 OCR Further Maths: Formula & Theorem Quick Reference | 8年级OCR进阶数学公式定理速查手册

📚 Year 8 OCR Further Maths: Formula & Theorem Quick Reference | 8年级OCR进阶数学公式定理速查手册

This quick reference handbook brings together all the essential formulae and theorems required for Year 8 OCR Further Mathematics. It is designed to support revision and problem-solving across number, algebra, geometry, statistics and probability.

本速查手册汇集了8年级OCR进阶数学所需的所有关键公式和定理，旨在辅助复习并解决涉及数、代数、几何、统计和概率的各类问题。

1. Number Facts and Fraction Operations | 数的性质与分数运算

Addition: a/b + c/d = (ad + bc) / bd. Find a common denominator first.

加法：a/b + c/d = (ad + bc) / bd。先通分再相加。

Subtraction: a/b - c/d = (ad - bc) / bd, again using a common denominator.

减法：a/b - c/d = (ad - bc) / bd，同样需要公分母。

Multiplication: c/d = (a × c) / (b × d). Multiply numerators and denominators straight across.

乘法：a/b × c/d = (a × c) / (b × d)。分子与分母直接相乘。

Division: a/b ÷ c/d = a/b × d/c. Multiply by the reciprocal of the divisor.

除法：a/b ÷ c/d = a/b × d/c。除以一个分数等于乘以它的倒数。

Fraction to decimal: divide the numerator by the denominator. e.g. 3/8 = 0.375.

分数转小数：用分子除以分母。例如 3/8 = 0.375。

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2. Indices Rules | 指数法则

Multiplication: a^m × aⁿ = a^m+n. Add the exponents when bases are the same.

乘法：a^m × aⁿ = a^m+n。底数相同时将指数相加。

Division: a^m ÷ aⁿ = a^m-n (a ≠ 0). Subtract the exponent of the denominator.

除法：a^m ÷ aⁿ = a^m-n（a ≠ 0）。用分子的指数减去分母的指数。

Power of a power: (a^m)ⁿ = a^m×n. Multiply the exponents.

幂的乘方：(a^m)ⁿ = a^m×n。指数相乘。

Zero index: a⁰ = 1, provided a ≠ 0.

零指数：a⁰ = 1，其中 a ≠ 0。

Negative index: a^-n = 1 / aⁿ. Reciprocal of the positive power.

负指数：a^-n = 1 / aⁿ。取正指数幂的倒数。

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3. Expanding and Factorising | 展开与因式分解

Distributive law: a(b + c) = ab + ac. Multiply each term inside the bracket.

分配律：a(b + c) = ab + ac。将括号外的项与括号内每一项相乘。

Difference of two squares: (a + b)(a - b) = a² - b².

平方差公式：(a + b)(a - b) = a² - b²。

Perfect square: (a + b)² = a² + 2ab + b²; (a - b)² = a² - 2ab + b².

完全平方：(a + b)² = a² + 2ab + b²；(a - b)² = a² - 2ab + b²。

Factorising by common factor: find the highest common factor (HCF) of all terms and reverse the distributive law.

提公因式法：找出所有项的最高公因式（HCF），倒用分配律进行分解。

Factorising quadratics: x² + bx + c = (x + p)(x + q) where p + q = b and pq = c.

二次三项式因式分解：x² + bx + c = (x + p)(x + q)，其中 p + q = b 且 pq = c。

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4. Solving Linear Equations and Inequalities | 解一元一次方程与不等式

General linear equation: ax + b = c → x = (c - b) / a. Perform inverse operations on both sides.

一元一次方程通解：ax + b = c → x = (c - b) / a。对方程两边进行逆运算求解。

Bracketed forms: expand first, then collect like terms, and isolate the variable.

含有括号的形式：先展开、再合并同类项，最后将未知数单独留在一边。

Inequality rules: if you multiply or divide both sides by a negative number, reverse the inequality sign. e.g. -2x < 4 ⇒ x > -2.

不等式规则：若在两边同时乘以或除以一个负数，必须反转不等号。例如 -2x < 4 ⇒ x > -2。

Compound inequalities: for a < b < c, solve each part while keeping the variable in the middle.

复合不等式：对于 a < b < c，分别求解各部分并保持变量在中间。

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5. Quadratic Formula and the Discriminant | 一元二次方程求根公式与判别式

Standard form: ax² + bx + c = 0, where a ≠ 0.

标准形式：ax² + bx + c = 0，其中 a ≠ 0。

Roots: x = [ -b ± √(b² - 4ac) ] / (2a)

根公式：x = [ -b ± √(b² - 4ac) ] / (2a)

Discriminant Δ = b² - 4ac. istinct real roots; if Δ = 0, one repeated real root; if Δ < 0, no real roots.

判别式 Δ = b² - 4ac。若 Δ > 0，有两个不等实根；若 Δ = 0，有一个重根；若 Δ < 0，无实根。

Completing the square shortcut: x² + bx = (x + b/2)² - (b/2)². This leads to the quadratic formula.

配方法速记：x² + bx = (x + b/2)² - (b/2)²。由此可导出求根公式。

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6. Pythagoras' Theorem and Distance | 勾股定理与距离公式

Pythagoras' theorem: In a right-angled triangle, c² = a² + b², where c is the hypotenuse.

勾股定理：在直角三角形中，斜边的平方等于两直角边的平方和，c² = a² + b²。

Finding a shorter side: a = √(c² - b²). Rearrange before taking the square root.

求直角边：a = √(c² - b²)。先移项再开平方。

Distance between two points (x&sb1;, y&sb1;) and (x&sb2;, y&sb2;): d = &radic;[(x&sb2; - x&sb1;)² + (y&sb2; - y&sb1;)²]. Derived from Pythagoras.

两点间距离公式：d = &radic;[(x&sb2; - x&sb1;)² + (y&sb2; - y&sb1;)²]。由勾股定理推导而来。

3D extension: for a cuboid, space diagonal = √(l² + w² + h²).

三维延伸：长方体的体对角线长 = √(l² + w² + h²)。

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7. Linear Functions and Equations of Lines | 一次函数与直线方程

Slope-intercept form: y = mx + c, where m = gradient, c = y-intercept.

斜截式：y = mx + c，其中 m 为斜率，c 为 y 轴截距。

Gradient between two points: m = (y₂ - y₁) / (x₂ - x₁).

两点间斜率：m = (y₂ - y₁) / (x₂ - x₁)。

Point-slop form: y - y₁ = m (x - x₁). Useful when you know a point and the gradient.

点斜式：y - y₁ = m (x - x₁)。适用于已知一点和斜率的情况。

Midpoint of line segment: M = ( (x₁+x₂)/2 , (y₁+y₂)/2 ).

线段中点：M = ( (x₁+x₂)/2 , (y₁+y₂)/2 )。

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8. Perimeter, Area and Volume | 周长、面积与体积

Rectangle: Perimeter = 2(l + w), Area = l × w.

矩形：周长 = 2(l + w)，面积 = l × w。

Triangle: Area = ½ × base × height. Perimeter = sum of three sides.

三角形：面积 = ½ × 底 × 高，周长为三边之和。

Parallelogram: Area = base × vertical height.

平行四边形：面积 = 底 × 垂直高。

Trapezium: Area = ½ (a + b) h, where a and b are parallel sides, h is height.

梯形：面积 = ½ (a + b)h，a, b 为平行边，h 为高。

Cube / cuboid: Volume = l × w × h, surface area = 2(lw + lh + wh).

立方体/长方体：体积 = l × w × h，表面积 = 2(lw + lh + wh)。

Prism: Volume = area of cross-section × length. Applies to all prisms.

棱柱：体积 = 横截面积 × 长度。适用于所有棱柱体。

Cylinder: Volume = πr²h, curved surface area = 2&pi;rh, total surface area = 2&pi;r(r + h).

圆柱体：体积 = πr²h，侧面积 = 2&pi;rh，全面积 = 2&pi;r(r + h)。

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9. Circle Geometry: Circumference, Area, Arc and Sector | 圆：周长、面积、弧长与扇形面积

Circumference: C = 2&pi;r or C = πd, where d = 2r.

圆周长：C = 2&pi;r 或 C = πd，其中 d = 2r。

Area of a circle: A = πr².

圆的面积：A = πr²。

Arc length: l = (&theta;/360&deg;) × 2πr. θ is the central angle in degrees.

弧长：l = (&theta;/360&deg;) × 2πr。θ 为圆心角度数。

Area of sector: A = (&theta;/360&deg;) × πr².

扇形面积：A = (&theta;/360&deg;) × πr²。

Perimeter of a sector: P

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