Math AA SL IA ideas: model, optimise and make a real decision
Test two models before choosing your exploration
Suppose you want to model how the temperature of a drink changes over time. Before committing to the title, collect a small pilot dataset and decide what each variable means. A linear fit may describe a short interval while an exponential model better represents cooling towards room temperature. Neither should be chosen simply because software reports an impressive fit.
- Plot observations with units and explain the measurement interval.
- Compare residuals: systematic curvature can reveal what a straight line misses.
- Reserve some observations to check the model outside the fitted sample.
- Explain the domain: a model that predicts cooling below the surrounding room temperature needs scrutiny.
A numerical check you can explain yourself
If a model predicts 42 °C and the observed temperature is 40 °C, the observed-minus-predicted residual is −2 °C. Calculate several residuals and look for a pattern, rather than discussing only one. These are illustrative values, not a ready-made IA dataset. The exploration should show your own reasoning and interpretation.
Compare the HL modelling ideas without assuming harder mathematics automatically produces a better exploration. Use the IA improvement checks and assessment-weighting example to connect the work to its purpose.
IA weighting and scoring criteria
The Math AA SL IA is a mathematical exploration. A strong SL exploration still needs clear mathematical communication, authentic choices, reflection and a precise mathematical decision, even when the methods are more focused than HL.
Clear aim, logical order, readable graphs/tables and a conclusion that answers the research question.
Define variables, use correct notation, label axes, state units and explain what each formula means.
Make the context yours: your phone, canteen, container, chain, ramp, measurements or design decision.
Compare models, test assumptions, discuss errors and explain what the answer means in the real situation.
Use course-level mathematics accurately; for HL, add derivation, proof, model comparison or numerical depth.
A strong Math AA SL IA should feel like a clean mathematical investigation, not a worksheet with a theme placed on top. The best SL topics use accessible mathematics very well: functions, modelling, optimisation, graphs, residuals, percentage error and interpretation. The goal is to choose a real situation where the mathematics helps you decide something practical.
This guide is based on the attached IA topic document. Each idea below has a research question, a model, observable data, calculations and a real conclusion. The point is not to copy the wording. Personalise the topic through your choices and reasoning: use your phone, your school canteen, your local cafe, or real containers you can measure. First-hand measurements can support the investigation, but theoretical explorations can also be valid.
What a strong IA in this subject shows
A high-scoring SL exploration usually does one thing very clearly. Start with a focused research question, then introduce the theory before the data. Show sample calculations, not only calculator output. If you fit more than one model, compare them using residuals, percentage error, or a reasoned discussion of shape. The final section should answer a decision question: best range, best dimensions, minimum number of servers, or a recommendation that follows from the mathematics.
Three go-to Math AA SL IA ideas
1. Optimising phone charging efficiency
This is a modern, personal IA because every student understands the problem: you have fifteen minutes before leaving, so what battery percentage range gives the most useful charge?
Research question: What battery percentage range gives the most usable battery life per minute of charging for my phone?
Charge the same phone with the same charger from about 10% to 100% and record the battery percentage every two or five minutes. The raw graph should not be perfectly linear, because most phones charge quickly at low percentages and slow down near the top. You can fit a linear, exponential or logistic-style model, then calculate interval efficiency for different charging windows. The IA becomes much stronger when it becomes an optimisation problem: best 15-minute charge, best stopping percentage, or the point where charging becomes inefficient.
IB topics used: functions, inverse functions, exponential models, optimisation, residuals
Data to collect: Battery percentage versus time, repeated under the same charger, temperature and phone-use conditions.
Diagram to include: Charging curve with several highlighted intervals, then a second graph of efficiency against starting percentage.
2. Optimising the shape of a drink can
This one looks simple, but it becomes powerful because you can measure real cans and then prove how much extra material each design uses compared with the mathematical optimum.
Research question: What cylinder dimensions minimise surface area for a fixed volume, and how close are real cans to this optimum?
Measure the radius and height of several real containers: a standard soda can, a slim energy drink can, a soup can, or a tube. For a fixed volume, rewrite the surface area of a cylinder as a function of radius only, differentiate it, and find the minimum. Then compare the theoretical optimum with actual product dimensions. The best reflection is not just that companies are inefficient. Explain why a real can may sacrifice minimum material for grip, branding, stacking, strength, shelf visibility or vending-machine compatibility.
IB topics used: calculus, surface area, volume, optimisation, percentage error
Data to collect: Measured radius, height and stated volume for real containers, followed by theoretical and actual surface area calculations.
Diagram to include: Cylinder diagram with radius and height, plus a surface-area curve showing the minimum point.
3. Optimising school canteen or cafe waiting time
This turns a normal lunch queue into a mathematical operations problem: how many servers are actually needed before opening another till is worth it?
Research question: How many servers are needed to reduce the average waiting time in a school canteen or cafe below a target value?
Observe a canteen, cafe or shop during a consistent peak period. Count arrivals per minute, record service times and note queue length at fixed intervals. Calculate the arrival rate and service rate, then compare the current system with one, two or three servers. The strongest version has a measurable target such as keeping average wait below five minutes. Reflect on assumptions: customers do not arrive evenly, orders differ, and service speed may change under pressure. This makes the IA realistic instead of pretending the queue is perfectly smooth.
IB topics used: rates, linear models, inequalities, modelling, optimisation
Data to collect: Arrivals per minute, service times, number of servers and queue length at fixed intervals.
Diagram to include: Timeline of arrivals and service completions, then queue length against time for different server numbers.
How to choose from these ideas
Choose the investigation where you can collect the cleanest data, not the one with the most dramatic title. A high-scoring IA usually has one clear independent variable, one meaningful dependent variable, enough repeats, a sensible model and an evaluation that admits what the data cannot prove. Before starting, confirm your exact subject guide, safety rules and teacher expectations.
Sources checked
- IB Mathematics in the DP
- IB Mathematics: Analysis and Approaches subject brief
- IB 2026 Mathematics examiner instructions
- IB assessment and exams
Editorial review
Reviewed against the cited official IB subject and assessment sources. This article provides planning examples, not examiner advice or ready-to-submit coursework.
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