📚 IGCSE Mathematics Teacher's Guide: 1000 Core Practice Questions | IGCSE 数学教师用书：1000 道核心练习题精讲
This bilingual teacher's guide accompanies the IGCSE Mathematics 1000-question core practice series. It summarises the key knowledge, common misconceptions and exam techniques that teachers should emphasise in every topic.
本双语教师用书配套 IGCSE 数学 1000 道核心练习题系列，总结了教师在每个专题中应强调的关键知识、常见误解和考试技巧。
1. Number and Sets | 数与集合
A secure number foundation is essential because almost every IGCSE question uses fractions, decimals, percentages, indices or surds. Teachers should check that students can switch confidently between equivalent forms.
扎实的数字基础至关重要，因为几乎每道 IGCSE 题目都会涉及分数、小数、百分数、指数或根式。教师应检查学生能否在不同等价形式之间自如转换。
aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1
The laws of indices are regularly tested in simplification questions. Students often forget that any non-zero number raised to the power zero equals 1, or they confuse multiplication and power rules.
指数法则在化简题中经常考查。学生常忘记任何非零数的零次幂等于 1，或者混淆乘法法则与幂的法则。
ample, 60 = 2² × 3 × 5 and 72 = 2³ × 3², so HCF = 2² × 3 = 12 and LCM = 2³ × 3² × 5 = 360.
求最大公因数 HCF 和最小公倍数 LCM 时，应使用质因数分解而不是列举倍数。例如 60 = 2² × 3 × 5，72 = 2³ × 3²，因此 HCF = 2² × 3 = 12，LCM = 2³ × 3² × 5 = 360。
Set notation appears in both core and extended papers: A ∪ B means union, A ∩ B means intersection, and A' means the complement of set A. Venn diagrams help visualise these relationships.
集合记号在核心和扩展试卷中都会出现：A ∪ B 表示并集，A ∩ B 表示交集，A' 表示集合 A 的补集。维恩图有助于直观展示这些关系。
2. Algebraic Manipulation | 代数运算
Expanding brackets, factorising quadratics and rearranging formulae are the backbone of Paper 2 and Paper 4. Students need fluency with positive and negative coefficients.
展开括号、二次因式分解和公式变形是 Paper 2 和 Paper 4 的支柱。学生需要熟练处理正负系数。
(a + b)² = a² + 2ab + b², (a - b)² = a² - 2ab + b², a² - b² = (a - b)(a + b)
b², which misses the 2ab term and loses marks even when the rest of the method is correct.
这些标准恒等式能加快展开和因式分解。常见的错误是写成 (a + b)² = a² + b²，漏掉了 2ab 项，即使方法其余部分正确也会失分。
Teachers should encourage students to expand and then refactorise to check their answers. For example, expanding (x + 3)(x - 2) gives x² + x - 6, and refactorising x² + x - 6 should return the same pair.
教师应鼓励学生先展开再重新因式分解来检查答案。例如展开 (x + 3)(x - 2) 得 x² + x - 6，重新因式分解 x² + x - 6 应得到相同的因式。
Rearranging formulae requires balancing operations in reverse order. If you need to isolate a variable, undo addition or subtraction first, then multiplication or division, and finally powers or roots.
公式变形需要按逆运算顺序进行平衡。如果要分离某个变量，应先去掉加减，再去掉乘除，最后处理乘方或开方。
3. Linear Equations and Inequalities | 线性方程与不等式
Linear equations often have unknowns on both sides, and students must collect like terms without losing negative signs. Solving an inequality follows the same steps, but the direction reverses when multiplying or dividing by a negative number.
线性方程常包含等式两侧都有未知数的情况，学生必须合并同类项且不能丢失负号。解不等式遵循相同步骤，但当乘以或除以负数时，不等号方向要改变。
ax + b = cx + d → x = (d - b)/(a - c), a ≠ c
Always substitute the solution back into the original equation to verify it. This takes only a few seconds and catches sign errors or arithmetic slips.
务必将解代回原方程进行验证。这只需几秒钟，却能发现符号错误或计算失误。
a closed circle for x ≤ 3 or x ≥ 3. The arrow shows all values that satisfy the condition.
在数轴上表示不等式时，严格不等式如 x < 3 或 x > 3 用空心圆圈，而 x ≤ 3 或 x ≥ 3 用实心圆圈。箭头表示所有满足条件的值。
For simultaneous linear equations, both substitution and elimination are acceptable. Encourage students to choose the method with the smaller arithmetic risk: if one equation already has a single variable, substitution is quick; otherwise elimination is usually safer.
对于联立线性方程，代入法和消元法均可接受。鼓励学生选择算术风险较小的方法：如果某个方程已经有一个单独的变量，代入法较快；否则消元法通常更稳妥。
4. Functions and Graphs | 函数与图像
Function notation f(x) tells students that the output depends on the input x. Composite functions such as fg(x) mean apply g first, then f. Inverse functions reverse the original mapping.
函数记号 f(x) 告诉学生输出取决于输入 x。复合函数如 fg(x) 表示先作用 g，再作用 f。反函数则反转原来的对应关系。
f(x) = ax + b, f⁻¹(x) = (x - b)/a
f(x), swap x and y, then solve for y. Many students forget to swap variables first and simply rearrange the original equation, which gives the wrong expression.
求反函数时，应写出 y = f(x)，交换 x 和 y，然后解出 y。许多学生忘记先交换变量而直接整理原式，结果得到错误表达式。
the y-intercept. Parallel lines have equal gradients, while perpendicular lines have gradients that multiply to -1.
直线图像的形式为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。平行线斜率相等，而垂直线的斜率乘积为 -1。
ped like a parabola. If a > 0 the parabola opens upwards, and if a < 0 it opens downwards. The turning point can be found by completing the square or using symmetry.
二次函数图像 y = ax² + bx + c 呈抛物线形。若 a > 0，抛物线开口向上；若 a < 0，开口向下。顶点可通过配方法或对称性求得。
5. Ratio, Proportion and Percentages | 比、比例与百分数
Ratio problems often link with direct and inverse proportion, percentage increase and decrease, and compound interest. Students should write ratios as fractions to set up equations clearly.
比值问题常与正比例、反比例、百分比增减和复利结合。学生应把比写成分数形式，以便清晰地建立方程。
Percentage change = (new - original)/original × 100%
For compound interest, the multiplier method is more efficient than repeated percentage calculations. The formula below gives the final amount after n years at an annual rate r%.
对于复利，乘法因子法比重复计算百分数更高效。以下公式给出年利率 r% 下经过 n 年后的最终金额。
A = P(1 + r/100)ⁿ
If a quantity decreases, the multiplier becomes (1 - r/100). A common mistake is adding or subtracting r instead of converting it to a decimal multiplier.
如果数量减少，乘法因子变为 (1 - r/100)。常见错误是直接加减 r，而不是把它转换为小数乘法因子。
Always find the constant k first using the given pair of values, then answer the question.
正比例关系为 y = kx；反比例关系为 y = k/x。务必先用已知的一对值求出常数 k，然后回答问题。
6. Geometry: Angles and Polygons | 几何：角与多边形
Angle facts, parallel lines, and polygon angle sums are tested regularly in IGCSE. Students must know corresponding angles, alternate angles and co-interior angles when a transversal crosses parallel lines.
角的基本性质、平行线和多边形内角和是 IGCSE 常考内容。学生必须掌握截线穿过平行线时的同位角、内错角和同旁内角。
Sum of interior angles = (n - 2) × 180°
The sum of exterior angles of any convex polygon is 360°. For a regular polygon, each exterior angle is 360°/n, and each interior angle is 180° - 360°/n.
任何凸多边形的外角和都是 360°。对于正多边形，每个外角为 360°/n，每个内角为 180° - 360°/n。
When solving angle problems, mark known angles on the diagram and state the reason for each step, such as angles on a straight line sum to 180° or vertically opposite angles are equal.
求解角度问题时，应在图上标出已知角，并说明每一步的理由，例如平角之和为 180° 或对顶角相等。
Students often confuse interior and exterior angles. A simple check: an interior angle of a convex polygon must be less than 180°, while an exterior angle is also less than 180° but usually much smaller in regular polygons.
学生常混淆内角和外角。简单判断：凸多边形的内角必须小于 180°，外角也小于 180°，但在正多边形中外角通常小得多。
7. Pythagoras and Trigonometry | 毕达哥拉斯与三角学
Right-angled triangle problems require Pythagoras' theorem and the sine, cosine and tangent ratios. Students must label the hypotenuse, opposite and adjacent sides correctly before choosing a ratio.
直角三角形问题需要勾股定理以及正弦、余弦和正切比。学生在选择比值之前，必须正确标出斜边、对边和邻边。
a² + b² = c²
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
The hypotenuse is always the
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