📚 IGCSE Maths Teacher's Guide: 450 Core Skills for Exam Success | IGCSE 数学教师用书：450 核心技能精讲
This teacher-facing guide brings together 450 essential skills and strategies for IGCSE Mathematics. It is designed to support classroom delivery, intervention and revision, with clear links to the Cambridge IGCSE 0580 and 0607 syllabuses.
本教师用书整合了 IGCSE 数学 450 个核心技能与策略，旨在支持课堂教学、分层辅导与复习备考，并与剑桥 IGCSE 0580 和 0607 大纲紧密衔接。
1. Number Operations and Estimation | 数字运算与估算
Directed numbers follow four sign rules: adding a negative is subtraction, subtracting a negative is addition, multiplying or dividing two negatives gives a positive, and multiplying or dividing one negative and one positive gives a negative.
有向数的运算法则有四条：加上负数等于减去其绝对值；减去负数等于加上其绝对值；两个负数相乘或相除结果为正；一个负数与一个正数相乘或相除结果为负。
imation questions usually require rounding values to one significant figure before calculating.
标准形式写作 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。估算题通常要求先把数值四舍五入到一位有效数字再计算。
- BIDMAS order: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
- BIDMAS 运算顺序：括号、指数、除法/乘法、加法/减法。
- Common error: writing 3.0 × 10² as 30²; the correct value is 300.
- 常见错误：把 3.0 × 10² 写成 30²；正确结果是 300。
(-a) × (-b) = ab and (-a) × b = -ab
2. Fractions, Decimals and Percentages | 分数、小数与百分数
To add or subtract fractions, first find a common denominator. To multiply fractions, multiply numerators and denominators directly; to divide by a fraction, multiply by its reciprocal.
分数加减时，先化为同分母。分数相乘时，分子乘分子、分母乘分母；分数相除时，乘以除数的倒数。
Percentage change is calculated as (new value - original value) ÷ original value × 100%. A multiplier of 1.15 represents a 15% increase, while 0.85 represents a 15% decrease.
百分比变化用（新值 - 原值）÷ 原值 × 100% 计算。乘数 1.15 表示增加 15%，乘数 0.85 表示减少 15%。
| Fraction 分数 | Decimal 小数 | Percentage 百分数 |
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/5 | 0.6 | 60% |
3. Ratio and Proportion | 比与比例
A ratio compares two or more quantities in the same units. To divide a quantity in a given ratio, find the total number of parts, then multiply the value of one part by each ratio term.
比用于比较两个或多个同类量。按给定比例分配时，先求出总份数，再计算每一份的值，最后分别乘以比中的各项。
tant k, so doubling x doubles y. Inverse proportion means y = k/x, so doubling x halves y.
正比例关系为 y = kx，其中 k 为常数，因此 x 翻倍时 y 也翻倍。反比例关系为 y = k/x，因此 x 翻倍时 y 减半。
Direct: y = kx | Inverse: y = k/x
- Simplify ratios by dividing all terms by their greatest common divisor.
- 化简比时，所有项同时除以它们的最大公约数。
- ales are ratios: 1 : 50 000 means 1 cm on the map represents 50 000 cm = 500 m in reality.
- 地图比例尺是比：1 : 50 000 表示地图上 1 cm 代表实际 50 000 cm，即 500 m。
4. Algebraic Expressions and Formulae | 代数表达式与公式
To expand brackets, multiply every term inside the bracket by the term outside. To factorise, find the greatest common factor and write it outside the bracket.
展开括号时，用括号外的项乘以括号内的每一项。因式分解时，找出最大公因式并将其提到括号外。
Two important identities are the difference of two squares and the perfect square expansion. Substitution replaces variables with given numbers, and the order of operations must be followed.
两个重要恒等式是平方差公式和完全平方展开公式。代入法用给定的数值替换变量，并严格遵循运算顺序。
(a + b)(a - b) = a² - b² | (a + b)² = a² + 2ab + b²
- Change the subject of a formula by reversing operations step by step.
- 变换公式主项时，按相反的运算步骤逐步移项。
- When factorising a quadratic like x² + 5x + 6, look for two numbers that multiply to 6 and add to 5, namely 2 and 3.
- 因式分解如 x² + 5x + 6 时，寻找乘积为 6、和为 5 的两个数，即 2 和 3。
5. Linear Equations | 线性方程
. To solve it, isolate the variable by using inverse operations: subtract b, then divide by a.
线性方程形如 ax + b = c。求解时，通过逆运算分离变量：先减去 b，再除以 a。
When variables appear on both sides, collect like terms on one side and constants on the other. Always check your solution by substituting it back into the original equation.
当方程两边都含有变量时，将含变量项移到一边，常数项移到另一边。解出后务必代入原方程检验。
ax + b = c ⇒ x = (c - b) / a
- Equations with fractions: multiply every term by the lowest common denominator first.
- 含分数的方程：先给每一项乘以最小公分母。
- Example: 3x + 4 = 19 ⇒ 3x = 15 ⇒ x = 5.
- 示例：3x + 4 = 19 ⇒ 3x = 15 ⇒ x = 5。
6. Inequalities | 不等式
ilar to solving an equation, but multiplying or dividing both sides by a negative number reverses the inequality sign.
不等式使用符号 <、>、≤ 和 ≥。解不等式与解方程类似，但当两边同乘或同除以一个负数时，不等号方向必须改变。
Solutions are often shown on a number line: an open circle means the endpoint is not included, while a closed circle means it is included.
解集常用数轴表示：空心圆表示不包含端点，实心圆表示包含端点。
-2x < 6 ⇒ x > -3
- Double inequality: solve all three parts at the same time.
- 双联不等式：同时对三个部分进行求解。
- Always write the final inequality with the variable on the left for clarity.
- 为清晰起见，最终结果通常把变量写在左边。
7. Straight-Line Graphs | 直线图像
m is the gradient and c is the y-intercept. The gradient measures steepness and direction.
直线方程通常写作 y = mx + c，其中 m 为斜率，c 为 y 轴截距。斜率描述直线的倾斜程度和方向。
To find the gradient between two points, divide the change in y by the change in x. Parallel lines have the same gradient.
求两点之间的斜率时，用 y 的变化量除以 x 的变化量。平行直线具有相同的斜率。
m = (y₂ - y₁) / (x₂ - x₁)
- Plot the y-intercept first, then use the gradient to find a second point.
- 先标出 y 轴截距，再利用斜率确定第二个点。
- A positive gradient slopes upward from left to right; a negative gradient slopes downward.
- 斜率为正时，直线从左到右上升；斜率为负时，直线从左到右下降。
8. Geometry: Angles and Polygons | 几何：角与多边形
Angles on a straight line add to 180°, angles around a point add to 360°, and vertically opposite angles are equal. These facts are used to find unknown angles.
平角为 180°，周角为 360°，对顶角相等。这些基本事实常用于求解未知角。
angles is (n - 2) × 180°. The sum of exterior angles of any convex polygon is always 360°.
对于 n 边形，内角和为 (n - 2) × 180°。任何凸多边形的外角和恒为 360°。
Interior angle sum = (n - 2) × 180° | Exterior angle sum = 360°
- A regular polygon has equal sides and equal angles.
- 正多边形各边相等、各角也相等。
- Each interior angle of a regular n-gon = (n - 2) × 180° / n.
- 正 n 边形每个内角为 (n - 2) × 180° / n。
9. Trigonometry in Right Triangles | 直角三角形中的三角学
Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. It is used to find a missing side when two sides are known.
勾股定理指出，在直角三角形中，斜边的平方等于两条直角边的平方和。已知两边时，可用它求第三边。
The three trigonometric ratios are sine, cosine and tangent. To choose the correct ratio, identify the given side and the side to be found relative to the given angle.
三个三角函数比是正弦、余弦和正切。选择正确比例时，要判断已知边和所求边相对于已知角的位置。
sin θ = opposite/hypotenuse | cos θ = adjacent/hypotenuse | tan θ = opposite/adjacent
- Label sides first: opposite, adjacent and hypotenuse.
- 先标注三条边：对边、邻边和斜边。
- Use inverse trigonometric functions to find an angle when two sides are known.
- 已知两边求角度时，使用反三角函数。
10. Statistics and Probability | 统计与概率
The mean is the sum of all values divided by the number of values. The median is the middle value when data are ordered, and the mode is the most frequent value.
平均数是所有数值之和除以数据个数。中位数是数据排序后位于中间的值，众数是出现次数最多的值。
Probability of an event is the number of favourable outcomes divided by the total number of equally likely outcomes. Probabilities range from 0 to 1.
事件的概率等于有利结果的数量除以所有等可能结果的总数。概率取值范围在 0 到 1 之间。
mean = Σx / n | P(A) = n(A) / n(S)
- Range = highest value - lowest value; it measures spread.
- 极差 = 最大值 - 最小值；它衡量数据的分散程度。
- For mutually exclusive events A and B, P(A or B) = P(A) + P(B).
- 对于互斥事件 A 和 B，P(A 或 B) = P(A) + P(B)。
Published by TutorHao | IGCSE Maths Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图