IB Mathematics: Paper 3 Preparation Guide and Exam Strategies | IB数学：Paper 3备考指南与答题策略

📚 IB Mathematics: Paper 3 Preparation Guide and Exam Strategies | IB数学：Paper 3备考指南与答题策略

Paper 3 is the most distinctive part of IB Mathematics Higher Level. In both Analysis and Approaches HL and Applications and Interpretation HL, it is a one-hour written paper made up of extended problem-solving questions that ask you to explore a mathematical idea rather than recall a standard recipe.

Paper 3 是IB数学高级别课程中最具特色的一份试卷。无论是分析与方法（AA）HL，还是应用与解释（AI）HL，Paper 3 都是一小时笔试，由拓展性问题组成，要求你探究某个数学思想，而不是机械套用标准题型。

1. Understanding Paper 3 | 认识 Paper 3

The examiners are not just checking whether you can get an answer. They want to see how you investigate, generalize, prove, and communicate. The questions are often structured as a sequence of parts: a simple observation first, then a pattern, then a general result, and finally an interpretation.

考官不仅检查你是否能得出答案，更希望看到你如何探究、推广、证明和表达。Paper 3 的题目通常按步骤展开：先是一个简单观察，然后是规律，接着是一般性结论，最后是实际解读。

Because Paper 3 is worth a large number of marks in a short time, you need a strategy. Rushing into algebra without reading the whole question is the most common mistake.

因为 Paper 3 在短时间内占据很高的分值，你需要一套策略。不读完整道题就急着做代数，是最常见的错误。

2. Command Terms and What They Ask | 指令词解读

Paper 3 is built around precise command terms. Your answer changes depending on whether the paper says “Show that”, “Find”, “Hence”, or “Sketch”.

Paper 3 围绕精确的指令词展开。题目是要求“证明”“求”“由此”还是“作图”，你的作答方式完全不同。

Command Term \| 指令词	What to Do \| 作答要求
Show that \| 证明	Give a full logical derivation. Do not just state the result.
Find \| 求	Write working and final answer; use exact form or 3 significant figures if required.
Hence \| 由此	Use the immediately previous result. Starting from scratch may not gain the full method mark.
Hence or otherwise \| 由此或其他方法	The previous result is the fastest route, but other valid methods are accepted.
Sketch \| 作图	Show axes, intercepts, asymptotes, turning points, and domain restrictions.
State \| 写出	No explanation is needed, but the answer must be clear and unambiguous.

When you see “Show that”, you are being asked to produce a mini-proof. When you see “Hence”, you should write “Using the result from part (a), …” in your solution.

看到“Show that”时，你其实是在完成一个微型证明。看到“Hence”时，你应该在解答中写下“利用（a）小题的结果……”这样的过渡。

3. Time Management for the One-Hour Paper | 一小时时间管理

Paper 3 usually contains two long questions, and each mark costs about one minute of exam time. If a question is worth 55 marks, you cannot afford to spend 20 minutes on one part that is worth only 2 marks.

Paper 3 通常包含两道长题，大约每一分对应一分钟考试时间。如果试卷总分是55分，你就不能在只值2分的小问上花20分钟。

Spend the first 5 minutes reading both questions and highlighting definitions, conditions, and “Show that” targets.

先用5分钟通读两道题，标出定义、条件和需要证明的目标。
Plan roughly 25 minutes per question, with extra time for the final interpretive part.

每道题大约分配25分钟，留出额外时间处理最后的解释性小问。
If a part is blocking you, write down a partial attempt, leave space, and return after finishing the later parts.

如果某个部分卡住了，写下部分尝试，留出空位，先完成后继小问再回来。

4. Read the Whole Question First | 先通读全题

Paper 3 questions are designed as a story. The final part often reveals why the earlier parts exist. If you read only part (a), you may miss a connection that would have told you exactly what method to use.

Paper 3 的题目是按故事线设计的。最后一问往往揭示了前面小问存在的意义。如果你只读（a），就可能错过一个能提示你使用什么方法的线索。

For example, if part (c) asks for “the limit of aₙ as n → ∞”, then part (a) is probably asking you to find a general formula for aₙ. Keep that destination in mind while you solve the earlier parts.

例如，如果（c）问“aₙ 当 n → ∞ 时的极限”，那么（a）很可能是在让你求出 aₙ 的通项公式。你在做前面小问时就要记住这个最终目标。

5. Identify the Underlying Mathematical Model | 识别潜在数学模型

Every Paper 3 question is built on one or two core ideas. For AA HL, the underlying model might be a sequence, a differential equation, a function family, or a proof by induction. For AI HL, it might be a graph model, a statistical distribution, a Markov chain, or a differential equation applied to a real-world situation.

每道 Paper 3 题目都围绕一到两个核心思想展开。AA HL 的模型可能是数列、微分方程、函数族或数学归纳法；AI HL 的模型可能是图模型、统计分布、马尔可夫链或应用于现实情境的微分方程。

Ask yourself: “What mathematical tool fits this context?” Write down the standard equations before doing calculations.

问自己：“什么数学工具适合这个情境？”先写下标准方程，再开始计算。
Check whether the context suggests exponential growth, a periodic function, a recurrence relation, or a proportional rate.

检查情境是否暗示指数增长、周期函数、递推关系或比例变化率。
In AI HL, state your modelling assumptions clearly, such as “the population is continuous” or “the transition probabilities are constant”.

在 AI HL 中，要清楚说明建模假设，例如“人口是连续的”或“转移概率恒定”。

6. Use Your Calculator Wisely | 合理使用计算器

Paper 3 allows a calculator for both AA HL and AI HL, but a calculator cannot do the thinking for you. Use it to explore patterns, check answers, and perform tedious numerical work.

AA HL 和 AI HL 的 Paper 3 都允许使用计算器，但计算器不能替你思考。用它来探索规律、检验答案、完成繁琐的数值计算。

Know your GDC functions thoroughly: equation solver, numerical integration, differentiation, matrix operations, statistics, graphing, and table mode. On AI HL, matrix powers and inferential statistics are especially important.

你要熟练掌握图形计算器（GDC）的功能：方程求解、数值积分、求导、矩阵运算、统计、绘图和表格功能。在 AI HL 中，矩阵幂和推断统计尤其重要。

Store intermediate values in the calculator memory instead of writing rounded decimals and re-typing them.

把中间结果存入计算器存储器，而不是写下四舍五入的小数再重新输入。
Use the graph to estimate roots, intersections, maxima, and minima before solving algebraically.

在代数求解前，先用图像估算零点、交点、最大值和最小值。
For “Show that” questions, the calculator can only confirm the result. You must still present a written proof.

对于“证明”题，计算器只能验证结果，你仍然需要写出书面证明。

7. Show Full Working for “Show That” Questions | “证明”题要写清完整过程

“Show that” questions are the heart of Paper 3. You cannot score full marks by stating the given answer. You need to show every key algebraic step, even if the jump seems obvious to you.

“证明”题是 Paper 3 的核心。你不能只把题末给出的结果抄一遍就得分。每一步关键代数变换都要写清楚，即使你觉得自己已经很明显。

For example, to show that ∫₀¹ x e⁻ˣ dx = 1 – 2/e, you should integrate by parts with u = x and dv = e⁻ˣ dx, then evaluate the boundary terms. Write down the method even if you are confident the answer is correct.

例如，要证明 ∫₀¹ x e⁻ˣ dx = 1 – 2/e，你应该设 u = x，dv = e⁻ˣ dx，用分部积分法，再代入上下限。即使你确定结果正确，也要写出方法过程。

State the method: “Integration by parts”, “Mathematical induction”, “Completing the square”, or “The squeeze theorem”.

先说明方法：“分部积分法”“数学归纳法”“配方法”或“夹逼定理”。
Do not use the answer in your proof. If you assume the statement is true, the proof is circular.

在证明过程中不要使用最终结论。如果预先假设命题成立，证明就成了循环论证。

8. Handle Generic Results and Parameters | 处理一般性结果与参数

Paper 3 often asks you to find a general expression in terms of n, to prove a formula for all positive integers, or to solve an equation with a parameter. The best way into these questions is to experiment with small cases first.

Paper 3 常要求你求出含 n 的通项、证明对所有正整数成立的公式，或求解带参数的方程。做这类题的切入点是先尝试小数值情况。

Suppose a question defines a sequence by u₁ = 1 and uₙ₊₁ = 2uₙ + 1. You can compute u₁ = 1, u₂ = 3, u₃ = 7, u₄ = 15, and conjecture uₙ = 2ⁿ – 1. Then prove the conjecture by induction

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Command Term \| 指令词	What to Do \| 作答要求
Show that \| 证明	Give a full logical derivation. Do not just state the result.
Find \| 求	Write working and final answer; use exact form or 3 significant figures if required.
Hence \| 由此	Use the immediately previous result. Starting from scratch may not gain the full method mark.
Hence or otherwise \| 由此或其他方法	The previous result is the fastest route, but other valid methods are accepted.
Sketch \| 作图	Show axes, intercepts, asymptotes, turning points, and domain restrictions.
State \| 写出	No explanation is needed, but the answer must be clear and unambiguous.

IB Mathematics: Paper 3 Preparation Guide and Exam Strategies | IB数学：Paper 3备考指南与答题策略