📚 IGCSE Mathematics Consolidation: Core Topics Masterclass | IGCSE 数学巩固：核心考点精讲
Welcome to this consolidation lesson for IGCSE Mathematics (Grade 2, Topic 10). This revision module gathers the most essential skills across number, algebra, geometry, trigonometry and statistics. Every section follows a clear pattern: key formula → worked example → common pitfall.
欢迎来到 IGCSE 数学巩固课（G2 主题 10）。本复习模块汇总了数、代数、几何、三角与统计中最核心的考点。每个小节均按”核心公式 → 例题演算 → 常见错误”的结构展开，帮助你系统查漏补缺。
1. Number Skills: Primes, HCF and LCM | 数的技能：素数、最大公约数与最小公倍数
Every integer greater than 1 can be written uniquely as a product of prime numbers; this is called prime factorisation. To find the Highest Common Factor (HCF), take the lowest power of each prime shared by all numbers. To find the Lowest Common Multiple (LCM), take the highest power of each prime that appears in any number.
每个大于 1 的整数都可以唯一地写成素数的乘积，这称为素因数分解。求最大公约数（HCF）时，取所有数共同素因数的最低次幂；求最小公倍数（LCM）时，取任一数中出现素因数的最高次幂。
Example: Find the HCF and LCM of 24 and 36.
24 = 2³ × 3, 36 = 2² × 3²
HCF = 2² × 3 = 12, LCM = 2³ × 3² = 72
例题：求 24 与 36 的最大公约数和最小公倍数。
24 = 2³ × 3，36 = 2² × 3²
HCF = 2² × 3 = 12，LCM = 2³ × 3² = 72
Common pitfall: Do not confuse HCF with LCM. The HCF is always ≤ the smaller number, while the LCM is always ≥ the larger number. Always verify your answers satisfy these bounds.
易错点：切勿混淆 HCF 与 LCM。HCF 一定小于等于较小数，LCM 一定大于等于较大数。算出结果后应立刻用这一性质检验。
2. Standard Form and Surds | 科学记数法与根式化简
Standard form expresses a number as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. This is essential for writing very large or very small quantities concisely.
科学记数法将数表示为 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。它是简洁书写极大或极小数值的核心工具。
Examples:
45,000 = 4.5 × 10⁴
0.00067 = 6.7 × 10⁻⁴
示例：
45,000 = 4.5 × 10⁴
0.00067 = 6.7 × 10⁻⁴
A surd is an irrational root such as √2 or √72. To simplify a surd, extract the largest square factor: √72 = √(36 × 2) = 6√2. To rationalise a denominator, multiply numerator and denominator by the surd: 1 ÷ √3 = √3 ÷ 3.
根式是无理根号式，如 √2 或 √72。化简根式时提取最大平方因数：√72 = √(36 × 2) = 6√2。有理化分母时，将分子分母同乘该根式：1 ÷ √3 = √3 ÷ 3。
Common pitfall: When multiplying in standard form, remember that indices add: (3 × 10⁵) × (2 × 10⁷) = 6 × 10¹². Check whether the coefficient stays below 10 after calculation.
易错点：科学记数法相乘时指数相加：(3 × 10⁵) × (2 × 10⁷) = 6 × 10¹²。计算后要检查系数是否仍然小于 10，否则需进位调整。
3. Algebra: Expanding and Factorising | 代数：展开与因式分解
Expanding brackets requires multiplying every term inside the bracket by the term outside: 3(x + 4) = 3x + 12. For double brackets, use the distributive rule twice: (x + 3)(x − 2) = x² + x − 6.
展开括号时，须将括号外的每一项乘以括号内的每一项：3(x + 4) = 3x + 12。对于双括号，运用两次分配律：(x + 3)(x − 2) = x² + x − 6。
The difference of two squares is a powerful shortcut: a² − b² = (a + b)(a − b). For example, x² − 25 = (x + 5)(x − 5).
平方差公式是一个高效技巧：a² − b² = (a + b)(a − b)。例如 x² − 25 = (x + 5)(x − 5)。
To factorise a quadratic x² + bx + c, find two numbers that multiply to c and add to b. For x² + 5x + 6, the numbers 2 and 3 work, giving (x + 2)(x + 3).
对二次式 x² + bx + c 因式分解时，找出两个数使它们的乘积为 c 且和为 b。对 x² + 5x + 6，2 和 3 满足条件，故得 (x + 2)(x + 3)。
Common pitfall: Signs matter! For x² − 7x + 12, both numbers must be negative: (x − 3)(x − 4). For x² − x − 6, use one positive and one negative: (x − 3)(x + 2).
易错点：正负号至关重要！x² − 7x + 12 中两个数均为负：(x − 3)(x − 4)；x² − x − 6 中一正一负：(x − 3)(x + 2)。
4. Linear Equations and Inequalities | 线性方程与不等式
Solving a linear equation means isolating the unknown using inverse operations. For example, solve 3x − 7 = 2x + 5: subtract 2x from both sides to get x − 7 = 5, then add 7: x = 12.
解线性方程就是通过逆运算将未知数分离出来。例如解 3x − 7 = 2x + 5：两边同时减去 2x 得 x − 7 = 5，再加 7 得 x = 12。
Linear inequalities follow the same rules as equations, with one crucial exception: if you multiply or divide by a negative number, the inequality sign reverses.
线性不等式的解法与方程基本一致，但有一个关键例外：当两边同乘或同除一个负数时，不等号必须反向。
Example: Solve −2x + 3 > 7.
Subtract 3: −2x > 4
Divide by −2 (reverse the sign): x < −2
例题：解 −2x + 3 > 7。
两边减 3：−2x > 4
同除以 −2（不等号反向）：x < −2
Common pitfall: When drawing a number line, use an open circle for < or >, and a closed circle for ≤ or ≥. Test a value from your solution region to confirm correctness.
易错点：在数轴上作图时，< 和 > 用空心圆，≤ 和 ≥ 用实心圆。代入解区域中的一个值检验结果是否正确。
5. Simultaneous Equations | 联立方程组
Simultaneous equations are solved by either the elimination method or the substitution method. In elimination, align the coefficients so that adding or subtracting the equations cancels one variable.
解联立方程组有两种常用方法：消元法和代入法。消元法通过调整系数使两式相加或相减后消去一个未知数。
Example: Solve 2x + y = 7 and x − y = 2.
Add the equations: 3x = 9, so x = 3.
Substitute x = 3 into x − y = 2: 3 − y = 2, so y = 1.
Solution: x = 3, y = 1.
例题：解方程组 2x + y = 7 与 x − y = 2。
两式相加：3x = 9，故 x = 3。
将 x = 3 代入 x − y = 2：3 − y = 2，得 y = 1。
解为：x = 3，y = 1。
Common pitfall: Always check your solution in the original equations. A sign error when subtracting is the most frequent mistake; write out each algebraic step clearly.
易错点：务必把解代回原方程验证。做减法时符号出错是最常见的失误；每一步代数运算都应清晰写出。
6. Quadratic Equations | 一元二次方程
A quadratic equation in the form ax² + bx + c = 0 can be solved by factorising, by completing the square, or by using the quadratic formula. If the expression factorises cleanly, that is usually the fastest route.
形如 ax² + bx + c = 0 的一元二次方程可通过因式分解、配方法或求根公式求解。若能顺利因式分解，该法通常最快。
Example: Solve x² − 5x + 6 = 0.
Factorise: (x − 2)(x − 3) = 0
Hence x = 2 or x = 3.
例题：解 x² − 5x + 6 = 0。
因式分解：(x − 2)(x − 3) = 0
因此 x = 2 或 x = 3。
When factorisation fails, use the quadratic formula:
当无法因式分解时，使用求根公式：
x = (−b ± √(b² − 4ac)) ÷ 2a
The discriminant Δ = b² − 4ac tells us the number of roots: if Δ > 0 there are two distinct real roots; if Δ = 0 there is one repeated root; if Δ < 0 there are no real roots.
判别式 Δ = b² − 4ac 决定根的个数：若 Δ > 0，有两个不同的实数根；若 Δ = 0，有一个重根；若 Δ < 0，无实数根。
Common pitfall: A quadratic equation must equal zero before factorising. For x² + 3x = 10, first rewrite as x² + 3x − 10 = 0, then factorise as (x + 5)(x − 2) = 0.
易错点：方程必须先化为零。对 x² + 3x = 10，应先改写为 x² + 3x − 10 = 0，再分解为 (x + 5)(x − 2) = 0。
7. Pythagoras’ Theorem | 勾股定理
Pythagoras’ theorem connects the three sides of a right-angled triangle: the square of the hypotenuse equals the sum of the squares of the other two sides.
勾股定理描述了直角三角形三边的关系：斜边的平方等于两直角边的平方和。
a² + b² = c²
Here c is the hypotenuse — the longest side, opposite the right angle. Example: if a = 3 and b = 4, then c² = 3² + 4² = 9 + 16 = 25, so c = 5.
其中 c 为斜边，即直角所对的最长边。例：若 a = 3，b = 4，则 c² = 3² + 4² = 9 + 16 = 25，故 c = 5。
To find a shorter side, rearrange the formula: a² = c² − b². For a triangle with c = 13 and b = 12, a² = 169 − 144 = 25, so a = 5.
求直角边时需变形公式：a² = c² − b²。对 c = 13、b = 12 的三角形，a² = 169 − 144 = 25，故 a = 5。
Common pitfall: The hypotenuse must always be identified correctly. Many students incorrectly substitute the wrong side into c. Sketch the triangle and mark the right angle first.
易错点：必须正确找出斜边。许多学生将错误的一边代入 c。建议先画图并标出直角，再确定斜边位置。
8. Trigonometry: SOH CAH TOA | 三角学：SOH CAH TOA
Trigonometry relates the angles of a right-angled triangle to the ratios of its sides. The three essential ratios are summarised by SOH CAH TOA.
三角学将直角三角形的角与边之比联系起来。三个基本比值由 SOH CAH TOA 概括。
sin θ = Opposite ÷ Hypotenuse
cos θ = Adjacent ÷ Hypotenuse
tan θ = Opposite ÷ Adjacent
sin θ = 对边 ÷ 斜边
cos θ = 邻边 ÷ 斜边
tan θ = 对边 ÷ 邻边
Example: In a right-angled triangle, the hypotenuse is 10 cm and the angle θ is 30°. To find the opposite side, use sin θ:
Opposite = 10 × sin 30° = 10 × 0.5 = 5 cm.
例题：直角三角形中斜边为 10 cm，角 θ = 30°。求对边时使用 sin θ：
对边 = 10 × sin 30° = 10 × 0.5 = 5 cm。
To find an angle, use the inverse functions: θ = sin⁻¹(Opposite ÷ Hypotenuse). Ensure your calculator is in degree mode unless the question specifies radians.
求角度时使用反三角函数：θ = sin⁻¹(对边 ÷ 斜边)。除非题目特别说明用弧度，否则请确认计算器处于角度（DEG）模式。
Common pitfall: The labels “opposite” and “adjacent” depend on which acute angle you are working with. Relabel the triangle each time you choose an angle.
易错点：“对边”和”邻边”的称呼取决于你所选的锐角。每次选定角度后都应重新标记三角形的各边。
9. Mensuration: Area and Volume | 几何测量：面积与体积
Mensuration requires memorising and applying standard formulae. For a circle, area A = πr² and circumference C = 2πr or πd. For a cylinder, volume V = πr²h and curved surface area = 2πrh.
几何测量需要熟记并运用标准公式。圆面积 A = πr²，周长 C = 2πr 或 πd。圆柱体积 V = πr²h，侧面积 = 2πrh。
For a triangular prism, volume = area of cross-section × length. If the triangular face has base 6 cm and height 4 cm, and the prism is 10 cm long, then V = ½ × 6 × 4 × 10 = 120 cm³.
三棱柱的体积 = 截面面积 × 棱长。若三角形截面底为 6 cm、高为 4 cm，棱长为 10 cm，则 V = ½ × 6 × 4 × 10 = 120 cm³。
Always keep consistent units when substituting into formulae. Convert all measurements to the same unit before calculating.
代入公式时务必保持单位一致。计算前先将所有测量数据换算为同一单位。
Common pitfall: Surface area requires every face to be counted. For a cylinder, remember to add the two circular ends: total surface area = 2πrh + 2πr².
易错点：表面积要算上每一个面。对于圆柱体，别忘了加上两个底面圆：总表面积 = 2πrh + 2πr²。
10. Linear Graphs | 线性函数图像
A straight line can be written in the form y = mx + c, where m is the gradient and c is the y-intercept. The gradient measures the steepness: it is the change in y divided by the change in x.
直线方程可写成 y = mx + c，其中 m 为斜率，c 为 y 轴截距。斜率衡量陡峭程度，等于 y 的变化量除以 x 的变化量。
Given two points (x₁, y₁) and (x₂, y₂), the gradient is:
已知两点 (x₁, y₁) 和 (x₂, y₂)，斜率为：
m = (y₂ − y₁) ÷ (x₂ − x₁)
A positive gradient slopes upward from left to right; a negative gradient slopes downward. A horizontal line has gradient 0; a vertical line has an undefined gradient.
斜率为正时直线从左向右上升；斜率为负时直线下降。水平线的斜率为 0；竖直线的斜率不存在。
Common pitfall: When calculating the gradient, keep the order of coordinates consistent. Mixing up which point you call (x₁, y₁) and (x₂, y₂) will not change the result, but a sign slip will.
易错点：计算斜率时坐标顺序要一致。交换 (x₁, y₁) 与 (x₂, y₂) 不会改变结果，但符号出错则会导致错误。
11. Statistics and Probability | 统计与概率
The three averages each summarise data in a different way. The mean is the total divided by the count, the median is the middle value after sorting, and the mode is the most frequent value. The range is the difference between the largest and smallest values.
三种平均数从不同角度概括数据。均值是总和除以个数；中位数是排序后的中间值；众数是出现次数最多的值。极差是最大值与最小值之差。
For the data set 4, 7, 7, 9, 13:
Mean = (4 + 7 + 7 + 9 + 13) ÷ 5 = 40 ÷ 5 = 8
Median = 7, Mode = 7, Range = 13 − 4 = 9
对数据集 4, 7, 7, 9, 13：
均值 = (4 + 7 + 7 + 9 + 13) ÷ 5 = 40 ÷ 5 = 8
中位数 = 7，众数 = 7，极差 = 13 − 4 = 9
Probability measures the chance of an event occurring, always between 0 and 1. For equally likely outcomes, P(event) = favourable outcomes ÷ total outcomes. If P(A) is the probability of an event, then P(not A) = 1 − P(A).
概率衡量事件发生的可能性，取值范围总在 0 到 1 之间。对等可能结果，P(事件) = 有利结果数 ÷ 总结果数。若 P(A) 为事件 A 的概率，则 P(非 A) = 1 − P(A)。
Common pitfall: For grouped data, the median is not simply the middle class but the value found by locating the middle position. For probability, always check that the sample space is complete before counting outcomes.
易错点：对分组数据，中位数不是简单取中间组，而是通过定位中间位置求得。对概率，计数前务必确认样本空间完整无遗漏。
12. Problem-Solving Strategy and Exam Tips | 解题策略与考试技巧
Across all topics, a consistent problem-solving approach will improve accuracy and speed. First, read the question twice and underline key data. Second, choose the appropriate formula or method. Third, solve step by step, writing every operation down. Finally, substitute your answer back to check it fits the original condition.
在所有主题中，稳定的解题流程能够同时提升准确率与速度。第一，读题两遍并划出关键数据。第二，选择恰当的公式或方法。第三，逐步求解，写出每一步运算。最后，将答案代回原题验证是否符合条件。
-
Show all working —
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导