📚 IGCSE Mathematics Problem-Solving Strategies | IGCSE数学解题策略
Success in IGCSE Mathematics is not only about memorising formulas; it is about choosing the right strategy when you face an unfamiliar question. This guide presents core problem-solving approaches that help you interpret questions, organise information, and reach accurate answers efficiently.
IGCSE数学考试的成功不仅在于记住公式，更在于面对陌生题目时能够选择正确的解题策略。本文将介绍核心的解题方法，帮助你理解题意、整理信息，并高效准确地得出答案。
1. Understand the Question | 理解题意
Before attempting any calculation, read the question two or three times. Identify what is given, what is unknown, and what condition connects them. Circle key words such as “find”, “show”, “estimate”, “not to scale”, and “give your answer correct to 3 significant figures”.
在开始任何计算之前，请把题目读两到三遍。明确已知量、未知量以及连接它们的条件。圈出关键词，例如“求”“证明”“估算”“未按比例绘制”“答案保留3位有效数字”等。
For example, if a question asks for the length of a side in a right-angled triangle, note whether you are given another side and an angle, or two sides. This tells you whether to use sine, cosine, tangent, or Pythagoras’ theorem.
例如，如果题目要求直角三角形某条边的长度，请注意题目给出的是另一条边和一个角，还是两条边。这决定了你应当使用正弦、余弦、正切还是勾股定理。
2. Work Backwards | 逆向推理
Some IGCSE questions are easier to solve from the desired result backwards to the known data. This is especially useful for questions involving reverse percentages, compound interest, or finding a missing quantity in a formula.
某些IGCSE题目从所求结果逆推到已知数据会更加容易。这对于涉及百分率还原、复利或公式中缺失量的问题尤其有效。
Suppose an item costs $92 after a 15% discount. To find the original price, let the original be x. The discounted price is 85% of x, so 0.85x = 92. Dividing both sides by 0.85 gives x = 108.24, to the nearest cent.
假设一件商品在打15%折扣后价格为92美元。设原价为x，则折后价为x的85%，因此0.85x = 92。两边除以0.85得x ≈ 108.24，精确到分。
0.85x = 92 → x = 92 ÷ 0.85 = 108.24
3. Draw a Diagram | 画图辅助
A clear diagram can transform a confusing geometry or trigonometry problem into a visual one. Mark every known length, angle, and right angle on the diagram. For bearings questions, always draw a north line at each point.
一张清晰的图可以把复杂的几何或三角函数问题转化为直观的视觉问题。在图上标出所有已知长度、角度和直角。对方位角问题，务必在每个点处画出指北线。
For example, when finding the shortest distance from a point to a line, drawing the perpendicular helps you see the right-angled triangle you need. Then apply Pythagoras or trigonometry directly.
例如，在求点到直线的最短距离时，画出垂线可以帮助你看出所需的直角三角形，然后直接运用勾股定理或三角函数。
Do not rely on the diagram in the exam paper if it says “not to scale”. Use it only as a sketch and transfer the given data accurately to your own drawing.
如果试卷上的图形标注“未按比例绘制”，不要依赖它。你只需把它当作草图，并将所有已知数据准确转画到自己的图中。
4. Identify Patterns | 寻找规律
Sequence questions often appear in IGCSE Paper 2 and Paper 4. Look for differences between consecutive terms, ratios, or second differences. Linear sequences have a constant first difference; quadratic sequences have a constant second difference.
数列题常见于IGCSE卷二和卷四。观察相邻项之差、比值或二阶差分。线性数列的一阶差分恒定；二次数列的二阶差分恒定。
For the sequence 5, 9, 13, 17, the first differences are +4, +4, +4. The nth term is 4n + 1. To check: when n = 1, 4(1) + 1 = 5; when n = 2, 4(2) + 1 = 9.
对于数列5, 9, 13, 17，一阶差分均为+4，因此第n项为4n + 1。检验：n = 1时，4(1) + 1 = 5；n = 2时，4(2) + 1 = 9。
For the sequence 2, 5, 10, 17, the first differences are 3, 5, 7 and the second differences are all 2. Therefore the nth term is quadratic and has the form an² + bn + c with a = 1.
对于数列2, 5, 10, 17，一阶差分为3, 5, 7，二阶差分均为2。因此第n项为二次式，形式为an² + bn + c，且a = 1。
5. Use Algebra to Generalise | 用代数推广
Algebra is a powerful tool for turning word problems into equations. Let the unknown be x, then translate each sentence into algebraic expressions. This helps you solve problems about ages, numbers, money, and mixtures.
代数是将文字题转化为方程的强大工具。设未知数为x，然后将每句话转化为代数表达式。这有助于解决年龄、数字、金额和混合类问题。
Consider: “The sum of three consecutive even numbers is 42.” Let the numbers be x, x + 2, x + 4. Then x + (x + 2) + (x + 4) = 42, so 3x + 6 = 42, giving x = 12. The numbers are 12, 14 and 16.
例如：“三个连续偶数的和为42。”设这三个数为x、x + 2、x + 4。则x + (x + 2) + (x + 4) = 42，所以3x + 6 = 42，解得x = 12。三个数分别为12、14和16。
x + (x + 2) + (x + 4) = 42 → 3x = 36 → x = 12
6. Break the Problem into Parts | 分而治之
Long multi-part IGCSE questions are designed to be solved step by step. The first part may ask you to find a value; the next part may use that value. Do not try to solve the entire problem at once.
IGCSE中较长的多部分题目是按步骤设计的。第一部分可能要求你求某个值，下一部分会用到这个值。不要试图一下子解决整个问题。
For a compound shape’s area, split it into rectangles, triangles, or semicircles. Calculate each area separately, then add or subtract as required. For a trapezium, use the formula Area = ½(a + b)h, where a and b are the parallel sides and h is the height.
对于组合图形的面积，可以将其拆分为矩形、三角形或半圆。分别计算每个面积，然后按需要相加或相减。对于梯形，使用面积公式：面积 = ½(a + b)h，其中a和b是平行边，h是高。
When working with cumulative frequency graphs, break the task into reading the median, the quartiles, and the interquartile range. Each part uses the same graph but a different reading skill.
在解决累积频率图问题时，可将任务分解为读取中位数、四分位数和四分位距。每一部分都使用同一张图，但考查不同的读数能力。
7. Eliminate Unnecessary Information | 排除干扰信息
IGCSE questions sometimes include extra numbers that are not needed. For example, when calculating the mean from a frequency table, ignore any numbers unrelated to the data set. Focus on the columns that give the value and frequency.
IGCSE题目有时会包含多余数字。例如，在通过频数表计算平均数时，忽略与数据集无关的数字，只关注给出数值和频数的列。
In a probability question, words like “with replacement” and “without replacement” are not extra; they change the denominator after each draw. Always underline these conditions.
在概率题中，“放回”和“不放回”并不是多余信息，它们会改变每次抽取后的分母。请务必在这些条件下方划线。
Convert all information into a table if possible. A table reveals which values are useful and which are just context.
如果可能，把所有信息整理成表格。表格可以揭示哪些数值有用，哪些只是背景信息。
8. Check Units and Scale | 检查单位与比例
Unit errors are a common cause of lost marks. If lengths are in metres but the answer is required in centimetres, you must convert correctly. Recall that 1 m = 100 cm and 1 cm = 10 mm.
单位错误是常见的失分原因。如果长度单位是米，但答案要求用厘米表示，则必须正确换算。记住1米 = 100厘米，1厘米 = 10毫米。
For area, the scale factor is squared. If two similar shapes have lengths in the ratio 2 : 5, their areas are in the ratio 4 : 25, and their volumes are in the ratio 8 : 125.
对于面积，比例因子要平方。如果两个相似图形的长度比为2 : 5，则它们的面积比为4 : 25，体积比为8 : 125。
Length ratio = 2 : 5 → Area ratio = 4 : 25 → Volume ratio = 8 : 125
Always write the correct unit in your final answer: cm, m², km/h, or $, depending on the question.
始终在最终答案中写出正确的单位：cm、m²、km/h或$，具体取决于题目要求。
9. Estimate Before Calculating | 先估算再计算
Estimation is a quick way to check whether your exact answer is reasonable. Round each number to one significant figure, do the arithmetic mentally, and compare it with your calculated answer.
估算是一种快速检验精确答案是否合理的方法。将每个数四舍五入到一位有效数字，进行心算，然后与你的计算结果比较。
For example, to estimate 48.7 × 3.14, round to 50 × 3 = 150. If your exact calculation gives 15.3 or 1530, you know you made a place-value or decimal-point error.
例如，估算48.7 × 3.14时，将其近似为50 × 3 = 150。如果你的精确计算结果为15.3或1530，说明你出现了数值位或小数点的错误。
Estimation is also useful for fractions and roots. Since √50 is approximately 7.07, any answer close to 7 is plausible, while an answer near 70 is clearly wrong.
估算对分数和根号同样适用。由于√50约等于7.07，因此接近7的答案较为合理，而接近70的答案显然有误。
10. Review and Verify Solutions | 检查与验证
After obtaining an answer, substitute it back into the original equation or condition. If it does not fit, re-read the problem and look for a sign error, a wrong formula, or a misread value.
得出答案后，请将其代回原方程或原条件进行验证。如果不符合，请重新阅读题目，检查符号错误、公式错误或数值误读。
For quadratic equations, you can check your two roots by adding them and multiplying them. If the roots are p and q, then p + q = -b/a and pq = c/a for ax² + bx + c = 0.
对于二次方程，可以通过两个根的和与积来检验。若根为p和q，则对于ax² + bx + c = 0，有p + q = -b/a，pq = c/a。
p + q = -b/a, pq = c/a
Finally, use the time remaining in the exam to review all your working. A small sign error can be corrected quickly if you notice it in time.
最后，请利用考试剩余时间检查所有解题过程。及时注意到一个小符号错误，往往能很快改正。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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