IB数学AA IA数学探究 选题灵感 写作详解
IB 数学内部评估(Mathematics IA,官方名称 Mathematical Exploration)占总分的 20%,是唯一一份完全由你自己控制分数的作业。许多同学把 IA 想得太难,迟迟不敢动笔,或者选题太泛导致 12 页里全是教科书复述,个人参与度不足直接被扣到 10 分以下。本文按 AA HL 与 AI SL 的真实评分标准,拆解选题原则、提供 10 个可直接上手的方向,并给出每步写作的结构模板。
The Mathematics Internal Assessment, officially called Mathematical Exploration, carries 20 percent of your final IB Maths grade, and it is the only piece of coursework whose score is fully under your control. Many students overestimate its difficulty and postpone starting, while others pick topics so broad that 12 pages become a textbook retelling with almost no personal engagement, losing marks straight down to single digits. This guide follows the real assessment criteria for AA HL and AI SL, explains how to choose a topic, offers ten directions you can start immediately, and gives a section by section structure for the write up.
1. 先看懂评分标准:五大 Criterion 决定你的写作策略
IB 数学 IA 的评分标准 A 到 E 五项:数学沟通(Communication)、数学表述(Mathematical presentation)、个人参与度(Personal engagement)、数学反思(Reflection)和数学应用(Use of mathematics)。2026 年 5 月大考起新版标准生效,但权重结构不变:前两项各占约 20%,个人参与度 20%,反思 20%,数学应用占最后约 20%。换句话说,选题本身只决定上限,而"你如何解释每一步"和"你对结果说了什么"才是分数的主要来源。
The IA is marked on five criteria, labelled A through E: Communication, Mathematical presentation, Personal engagement, Reflection, and Use of mathematics. The new criteria released for the May 2026 examination session keep the same weight balance: the first two are worth about 20 percent each, Personal engagement 20 percent, Reflection 20 percent, and the final criterion roughly 20 percent. In other words, the topic only sets your ceiling, while how you explain each step and what you say about your results drive most of the marks.
这意味着两件事。第一,AA HL 学生不必追求超出大纲的大学数学,把 SL 与 HL 内容用对、用透,配上真实数据与个人决策,完全可以拿 17 到 19 分。第二,AI SL 学生不要选纯微积分题目硬写,统计与建模方向更能发挥计算器与数据结构优势。选方向之前,先问自己:这个题目能让我展示大纲内的至少两种数学工具吗?能让我做出自己的选择吗?
Two implications follow. First, AA HL students do not need university level mathematics; using SL and HL content correctly and thoroughly, supported by genuine data and your own decisions, is enough to reach 17 to 19 out of 20. Second, AI SL students should avoid forcing pure calculus topics and instead choose statistics or modelling directions where the calculator and data handling skills give a real edge. Before choosing, ask yourself two questions: can this topic show at least two tools from the syllabus, and does it leave room for choices that are genuinely yours?
2. 选题三原则:聚焦、可量化、有个人决策点
第一条原则是聚焦。官方建议 12 到 20 页,你只有大约 3000 字的空间,所以一个题目只能容纳一个核心问题。把"研究彩票概率"收窄成"比较乐透与即开型彩票的期望回报",把"分析股市"收窄成"用移动平均与指数平滑预测一只股票的 30 日走势"。范围越窄,越容易做到有深度。
The first principle is focus. With 12 to 20 pages and roughly 3000 words available, one exploration can only carry one core question. Narrow a topic like lotteries in general to a comparison of expected returns between a draw game and a scratch card game, or narrow stock market analysis to forecasting one stock over 30 days using moving averages and exponential smoothing. A narrower scope makes depth achievable.
第二条原则是可量化:你的问题必须能落到数字、函数、概率或数据上,而不是纯文字推理。第三条原则是必须有个人决策点:数据由你自己收集,参数由你自己选择,方法 A 与 B 由你自己比较并给出取舍理由。评分官看个人参与度时,找的正是这些"为什么选它、为什么改它"的痕迹。
The second principle is measurability: your question must reduce to numbers, functions, probabilities, or data rather than verbal reasoning alone. The third is that your exploration must contain personal decision points: data you collected yourself, parameters you chose, and a comparison of method A against method B with reasons for your trade off. When examiners assess Personal engagement, they look exactly for traces of why you chose it and why you changed it.
3. 十大选题方向:按 AA 与 AI 适配度排列
方向一,复利与分期付款(AA SL 数列与级数)。研究一笔房贷在不同利率与期限下的还款总额,推导等额本息公式 S = P r (1+r)^n / ((1+r)^n - 1),再用真实银行利率代入计算。个人决策点:比较等额本息与等额本金两种还款方式哪个更优,讨论提前还款的临界点。这个题目几乎必拿基础分,因为推导链条完整、图表直观。
Direction one is compound interest and instalment loans, using the sequences and series topic from AA SL. Model a mortgage under different rates and terms, derive the equal repayment formula S = P r (1+r)^n divided by ((1+r)^n - 1), then substitute a real bank rate. Your decision point is comparing equal instalment with equal principal repayment, and finding the break even month for early repayment. This topic reliably earns base marks because the derivation chain is complete and graphs are intuitive.
方向二,Logistic 人口增长模型(AA HL 微分方程或 AI HL 建模)。用 dP/dt = kP(1 - P/L) 拟合一个城市或国家的人口数据,先用最小二乘或增长率线性化估计 k 与 L,再与纯指数模型对比残差。个人参与:解释为什么疫情期间模型失效,或把模型推广到两种群竞争。注意 AA 只需解出 Logistic 方程并讨论平衡点,不必引入大学内容。
Direction two is the logistic population model, linking to differential equations in AA HL or modelling in AI HL. Fit dP/dt = kP(1 - P/L) to population data of a city or country, estimate k and L by linearising growth rates or least squares, then compare residuals against a pure exponential model. Personal engagement comes from explaining why the model broke during a pandemic or extending it to two competing species. AA students only need to solve the logistic equation and discuss equilibrium points; no university material is required.
方向三,贝叶斯定理与医学检测(AI SL 概率)。取一种真实疾病的患病率与试剂敏感度、特异度,计算阳性结果下的后验患病率 P(D|+) = P(+|D)P(D) / P(+)。大多数人会惊讶:敏感度 95% 的检测,在患病率 1% 时阳性者真正患病的概率不到 16%。个人参与点:用树形图重算,讨论二次检测的价值,并批判媒体对"准确率"的误读。这是 AI 方向高性价比选题,计算量小、反思空间大。
Direction three is Bayes theorem in medical testing, part of AI SL probability. Take a real disease prevalence plus the sensitivity and specificity of a screening test, then compute the posterior probability of disease given a positive result, P(D|+) = P(+|D)P(D) divided by P(+). Most people are shocked that a test with 95 percent sensitivity, applied where prevalence is 1 percent, leaves a positive patient with under 16 percent chance of actually being ill. Your engagement points include recomputing with a tree diagram, evaluating the value of a second test, and criticising how the media reports accuracy figures. This is a high value AI topic with light computation and wide room for reflection.
方向四,赌徒破产问题与随机游走(AA HL 或 AI HL 概率)。一位玩家每次押注 1 元,赢的概率 p,问他从本金 a 到目标 b 的破产概率。推导差商方程并解出 p = 0.5 时破产概率为 1 - a/b。用 Excel 或计算器模拟 1 万局验证理论值。这个题目把递推、概率生成函数和模拟三种工具串起来,是展示数学深度的经典选择,但务必控制篇幅,只推导一维情形。
Direction four is the gambler ruin problem and random walks, suitable for AA HL or AI HL probability. A player bets one unit each round, wins with probability p, and we ask the ruin probability when starting from capital a and aiming for b. Derive the difference equation and solve it, obtaining ruin probability 1 - a/b when p equals 0.5. Simulate ten thousand games in Excel or on your calculator to verify the theoretical value. The topic chains together recurrence, probability generating functions, and simulation, a classic way to show depth, but limit the write up to the one dimensional case.
方向五,停车问题或电梯等待的优化(AA SL 函数与微积分)。把"在一条街上找车位"建模成概率密度,或用排队论近似超市收银台等待时间。个人参与:自己计时收集 50 组数据,用指数分布拟合到达间隔,再算平均队长。微积分优化与统计拟合同时出现,能覆盖 Criterion E 对"使用大纲外但恰当工具"的期待。
Direction five is an optimisation problem such as parking or waiting for an elevator, using functions and calculus from AA SL. Model finding a parking space on a street as a probability density, or approximate supermarket checkout waiting with queueing ideas. Engage personally by timing 50 arrivals yourself, fitting an exponential distribution to the intervals, then computing the mean queue length. Combining calculus optimisation with statistical fitting satisfies Criterion E's expectation of appropriate tools used correctly.
方向六,曲线长度与分形边界(AA HL 积分,或 AI SL 几何)。科赫雪花 Koch snowflake 是最稳妥的分形选题:每轮边长变为 1/3,数量乘 4,周长每轮乘 4/3 趋于无穷,而面积收敛于有限值 8/5 倍初始三角形面积。用等比级数分别求周长与面积,再讨论"海岸线悖论"。图形漂亮、推导简洁,AA HL 可用极限与级数严格化,AI 学生可做数值逼近。
Direction six covers arc length and fractal boundaries, using integration in AA HL or geometry in AI SL. The Koch snowflake is the safest fractal choice: each iteration divides every side by 3 and multiplies the number of sides by 4, so the perimeter grows by 4/3 each round towards infinity while the enclosed area converges to 8/5 of the initial triangle. Use geometric series for both perimeter and area, then discuss the coastline paradox. The figures are beautiful and the derivation compact; AA HL students can formalise with limits and series while AI students can work numerically.
方向七,体育数据中的相关与回归(AI SL 统计,AA SL 亦可)。选一个可下载的真实数据集,例如 F1 排位赛速度与正赛名次、NBA 三分命中率与胜率,或英超球队控球率与积分。先画散点图,计算皮尔逊相关系数,做最小二乘回归并检验残差是否随 x 变大而变大(异方差)。个人参与:解释为什么相关不等于因果,并讨论遗漏变量,例如球队预算。避免选 r 接近 1 的"太完美"数据,那反而没有讨论空间。
Direction seven analyses correlation and regression in real sport data, part of AI SL statistics and acceptable in AA SL. Download an open dataset such as F1 qualifying speed against race position, NBA three point percentage against win rate, or Premier League possession against points. Plot a scatter diagram, compute the Pearson correlation coefficient, run a least squares regression, and check whether residuals widen as x grows, a sign of heteroscedasticity. Engage personally by explaining why correlation is not causation and discussing omitted variables such as club budget. Avoid datasets with r close to 1; perfect data leaves nothing to discuss.
方向八,赌场游戏期望与策略(AI SL,离散概率分布)。以轮盘赌 Roulette 为例:欧洲盘 37 格,押单个数字的期望回报为 36/37 乘赌注,长期每注亏 2.7%。比较马丁格尔加倍下注策略的破产风险与小额稳定收益,用几何分布算连输 k 次的概率,说明"必胜策略"在有限本金下必破产。话题吸引力强,写作时保持数学严谨,不写赌博诱导内容即可。
Direction eight looks at expected value and strategy in casino games, using discrete distributions in AI SL. Take European roulette with 37 slots: betting a single number pays 36/37 of the stake in expectation, a long run loss of 2.7 percent per bet. Compare the Martingale doubling strategy's ruin risk against its small steady gains, compute the probability of k consecutive losses with the geometric distribution, and show that any guaranteed strategy collapses under finite capital. The topic is engaging; keep the mathematics rigorous and avoid promoting gambling.
方向九,音乐与数学:十二平均律(AA SL 指数与对数)。音高每升八度频率翻倍,十二平均律把八度分成 12 个等比半音,因此每个半音的频率比是 2 的 12 次方根。用等比数列验证钢琴调音,计算纯律五度(3:2)与平均律五度的微小偏差(约 2 音分),并解释为什么巴赫的十二平均律作品成为可能。适合有乐器背景的学生,个人参与度天然高。
Direction nine connects music and mathematics through equal temperament, using exponentials and logarithms from AA SL. Each octave doubles frequency, and equal temperament splits the octave into 12 geometrically equal semitones, so every semitone ratio is the twelfth root of 2. Verify piano tuning with a geometric sequence, compute the small deviation between a pure fifth of ratio 3:2 and the equal tempered fifth, about 2 cents, and explain why Bach's Well Tempered Clavier became possible. Students with a music background earn Personal engagement marks almost automatically here.
方向十,用数列分析社交媒体传播(AI SL,或 AA 数列)。假设一条帖子的转发满足 S(t+1) = S(t) + a S(t)(1 - S(t)/N) 的离散 Logistic 增长,用真实转发时间序列估计参数,比较与指数增长的拟合。或研究"裂变海报"的几何级数转发假设为何在现实中失败。数据收集方便(自己发一条测试帖记录每小时转发),且能自然引出模型局限性讨论。
Direction ten models social media spread with sequences, suitable for AI SL or AA sequences. Suppose shares follow the discrete logistic growth S(t+1) = S(t) + a S(t)(1 - S(t)/N), estimate parameters from a real share time series, and compare the fit against exponential growth. Alternatively examine why the geometric series assumption behind viral referral campaigns fails in practice. Data is easy to collect, for example posting your own test content and logging hourly shares, and the topic naturally leads to a discussion of model limitations.
4. 写作结构:六个部分的分工与篇幅
推荐结构依次是:引言(半页到一页),说明研究问题、动机与个人兴趣;数学背景(一页到两页),只写解题必需的定义与公式,不抄课本;主体探索(五到八页),按"建立模型、代入数据、计算结果、图形展示"循环推进;结果与讨论(一页),报告关键数值并解释其含义;反思(半页到一页),指出模型的假设、误差来源与可改进处;参考文献与附录(不计分但必须)。总页数控制在 12 到 18 页之间,图表占页数不等于字数。
A proven structure runs as follows. Introduction, half to one page, states the research question, motivation, and personal interest. Mathematical background, one to two pages, includes only definitions and formulas needed for the solution, not textbook copying. The main exploration, five to eight pages, cycles through building a model, substituting data, computing results, and presenting graphs. Results and discussion, one page, reports key numbers and interprets them. Reflection, half to one page, lists assumptions, error sources, and possible improvements. References and appendix are ungraded but compulsory. Keep the total between 12 and 18 pages, remembering that graphs consume pages without adding words.
公式呈现建议:手写或 LaTeX 生成的公式截图插入,保证清晰;每个公式第一次出现时解释每个符号含义。图表必须有标题、坐标轴标签与一句"图中可见……"的解读,评分官不会替你做分析。
For formulas, insert clear screenshots from handwriting or LaTeX, and explain every symbol the first time it appears. Every graph needs a title, labelled axes, and at least one sentence reading what the figure shows; examiners will not analyse it for you.
5. 常见失分点与规避方法
失分点一:整篇只有计算没有解释。规避方法:每三步计算之间插入一句中文或英文的"这一步在做什么、为什么这样做"。失分点二:引用网络资料不给来源。规避方法:数据与公式后直接注明网站与访问日期,IB 对学术诚信抽查严格,查重系统会标记未标注的复制段落。失分点三:结论只说"模型符合数据"。规避方法:给量化结论,例如"平均绝对百分比误差为 4.2%,在 10 个观测中 8 个落在 95% 预测区间内"。
Common mark loss comes from computation without explanation; fix it by inserting, every few steps, a sentence stating what this step does and why. Uncited web sources cause integrity flags; note the website and access date right next to each dataset and formula. Conclusions that merely say the model fits data earn nothing; instead report numbers such as a mean absolute percentage error of 4.2 percent with 8 of 10 observations inside the 95 percent prediction interval.
失分点四:选题超出大纲且不会用。用大学内容但推导错误比不用更糟,IB 官方立场是鼓励超出大纲的数学,但必须完全正确。失分点五:个人参与造假。不要编造"我测量了 100 次",评分官与审核员会核对数据合理性,收据式描述如"我在 3 个雨天与 3 个晴天各记录 15 辆公交车的到站时间"更可信。失分点六:AA HL 学生使用 AI 工具直接生成整篇文字,反思部分空泛。反思应写你尝试过什么方法失败、如何调整,这是 AI 无法替你诚实完成的部分。
Using university mathematics with a derivation error is worse than not using it; the IB encourages going beyond the syllabus only when the mathematics is fully correct. Never fabricate personal data such as claiming 100 measurements; examiners cross check plausibility, and receipt style descriptions like recording arrival times of 15 buses on 3 rainy and 3 sunny days read as genuine. Finally, AA HL students should not let AI tools write the whole text, because the reflection section must honestly describe methods you tried, failed, and adjusted, which is exactly what an AI cannot truthfully produce for you.
6. 动笔前检查清单
动笔前用这份清单自检:研究问题是否是一个能用数学回答的问句,而不是一个主题词;是否列得出题目用到的大纲内工具清单,至少两种;数据是自己收集或可追溯来源的吗;是否存在至少三处个人决策点;目标篇幅 12 到 18 页有分配计划吗;草稿中的每个公式都能解释符号吗;有没有为结论准备量化误差指标。七项全过,再开始写引言。记住:选题花一周思考不亏,写完后留三天重读,专找"只有计算没有解释"的段落补话。
Run this checklist before writing: is the research question a question answerable by mathematics rather than a topic label; can you list at least two syllabus tools the project uses; is the data self collected or traceable; are there at least three personal decision points; do you have a page budget for 12 to 18 pages; can you explain every symbol in your draft formulas; and do you have a quantitative error measure ready for the conclusion. Only when all seven pass should you start the introduction. Spending a week on topic selection is never wasted, and after drafting, leave three days before rereading, hunting specifically for stretches of bare computation that need explanation.
如果看完十个方向仍不确定,最稳妥的做法是从"方向一复利"或"方向三贝叶斯"起步,这两个题目资料充足、推导确定、反思空间大,足够支撑一篇 16 分以上的 IA。先完成一篇,再考虑是否值得换题重做。祝你的数学探究顺利。
If the ten directions still leave you uncertain, the safest move is to start with direction one on compound interest or direction three on Bayes theorem; both have abundant resources, deterministic derivations, and wide reflection space, enough to support an IA above 16 out of 20. Finish one complete draft first, then decide whether a different topic is worth the rewrite. Good luck with your mathematical exploration.
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