IB Maths IA: Exploration Topics and Criteria

The exploration is marked out of 20 and is worth 20% of your final Mathematics grade, at both SL and HL, in both Analysis and Approaches and Applications and Interpretation. Those 20 marks are not one global impression. They are five separate accounts, and each one is opened or closed by specific evidence in your document.

Criterion A, Presentation, out of 4.

Rewards a coherent, organised, concise piece. Coherent means the reader can follow your line of argument without jumping back. Organised means there is a visible structure. Concise means nothing is there for length: a five page appendix of raw spreadsheet output actively costs you here.

Criterion B, Mathematical communication, out of 4.

Rewards correct and consistent notation, terminology and forms of representation, plus defined variables and deductive reasoning. This is the easiest criterion to lose marks on and the easiest to fix.

Criterion C, Personal engagement, out of 3.

The bands run from some, to significant, to outstanding engagement. It is evidenced by what you did, not by what you say you felt.

Criterion D, Reflection, out of 3.

The bands run from limited, to meaningful, to critical reflection. Critical means you judge the mathematics, not the experience of writing it up.

Criterion E, Use of mathematics, out of 6.

The biggest single block. The mathematics must be relevant, commensurate with the level of your course, correct, and used with understanding. The top band asks for precision, sophistication and rigour.

The practical consequence: before you write a word, put the five criteria in a table down one side of a page and, next to each, write the exact page of your draft where the examiner will find the evidence. Any criterion with an empty cell is a criterion you are currently scoring zero on. Our guide to the IB internal assessment covers how this logic applies across the other Diploma subjects.

Most low scoring explorations were decided at the topic stage, not at the writing stage. A topic is viable only if it survives four questions.

Does it contain a question, not just a subject?

"Fractals" is a subject. "How well does the box counting dimension estimate the fractal dimension of the Lombardy coastline of Lake Como from map data?" is a question. Criteria A and D need a question, because coherence and reflection are both measured against an aim.

Is the mathematics commensurate with your course and level?

This is the single most expensive mistake. For AA HL, the syllabus sits around calculus, complex numbers, vectors, proof and series, so an exploration that stops at fitting a straight line to ten data points cannot reach the top of Criterion E, however beautifully written. For AI SL, statistical modelling, chi squared tests and applied optimisation are exactly the right register, and importing unexplained partial derivatives from the internet will lose marks rather than gain them, because you cannot demonstrate understanding of what you have borrowed. Check your course content honestly against the AA HL syllabus guide before committing.

Can you get the data or generate the examples yourself?

Explorations that depend on a dataset that turns out not to exist die in week four. If you use secondary data, cite it and say why it is fit for purpose.

Is there something for you to decide?

Criterion C rewards choices: choosing your own model, collecting your own measurements, testing a variant nobody suggested, applying a method to a context you personally care about.

Topics that reliably work include modelling a real decay or growth process you measured, optimising a genuine physical shape, comparing two forecasting models on the same data, deriving a result that is normally quoted to you, and applying a syllabus technique to a context outside mathematics such as music, sport or medicine. Topics that reliably fail include a history of a famous theorem, a proof copied from a textbook, and anything about the Monty Hall problem or the golden ratio, which examiners have seen thousands of times and which leave you no room to make decisions.

Criteria A and B are worth 8 of the 20 marks, they are largely mechanical, and they are the marks students most often leave on the table. A structure that works, section by section:

• Introduction, about one page. The question, why you chose it, and what you will do. State the aim in one sentence that a non specialist could repeat.
• Background mathematics, one to two pages. Only the theory you will actually use, with variables defined at the point of first use.
• The main development, six to ten pages. Derivation, data, model, calculation, results.
• Validation and extension, two to four pages. Testing the model, changing an assumption, pushing the method further.
• Conclusion, half a page. Answer the question you asked. Not a summary of the process.
• Bibliography and short appendix. Appendices hold raw data, never the argument.

On notation, the recurring losses are the same every year. Calculator and spreadsheet notation costs marks: write $2^{x}$ and not 2^x, write $3 \times 10^{5}$ and not 3E5, write $xy$ and not x*y. Use $=$ only between things that are equal, so never chain $\frac{dy}{dx} = 2x = 4$ when $x = 2$. Define every symbol: $T$ is the temperature in degrees Celsius, $t$ is the time in minutes since pouring. Keep one letter for one meaning throughout. Label every graph with quantity and unit on both axes, and give every figure a number you refer to in the text.

Graphical display calculator and software output is expected in a modern exploration, but a pasted screenshot is not communication. Every screenshot needs a caption saying what it shows and a sentence in the body saying what you concluded from it. If you delete a screenshot and the argument still stands, the screenshot was decoration and Criterion A will treat it as such.

Here is a complete miniature of what a top band section looks like. The aim: does Newton's law of cooling predict the temperature of a cup of coffee in my kitchen, and where does it fail?

Step 1, state the model and derive the solution.

Do not quote the exponential formula. Deriving it is exactly where Criterion B awards deductive reasoning, in the same way a paper question awards the method rather than the final number. Newton's law of cooling assumes the rate of cooling is proportional to the excess temperature over the surroundings:

$$\frac{dT}{dt} = -k(T - T_s)$$

where $T$ is the coffee temperature in degrees Celsius, $T_s$ the constant room temperature, $t$ the time in minutes and $k > 0$ a cooling constant with units of $\text{min}^{-1}$. Separating variables gives $\int \frac{1}{T - T_s} \, dT = \int -k \, dt$, so $\ln|T - T_s| = -kt + C$, and applying $T(0) = T_0$ gives

$$T(t) = T_s + (T_0 - T_s)e^{-kt}$$

Step 2, turn the model into something you can test.

Taking logs of the excess temperature linearises the model: $\ln(T - T_s) = \ln(T_0 - T_s) - kt$. So a plot of $\ln(T - T_s)$ against $t$ should be a straight line of gradient $-k$.

Step 3, use the measurements.

With $T_s = 22$, $T_0 = 90$ and a measured $T = 68$ at $t = 10$:

$$-k = \frac{\ln(68 - 22) - \ln(90 - 22)}{10 - 0} = \frac{3.829 - 4.220}{10} = -0.0391$$

so $k \approx 0.0391 \ \text{min}^{-1}$.

Step 4, predict and compare.

At $t = 20$ the model gives $T = 22 + 68e^{-0.0391 \times 20} = 22 + 68(0.4575) \approx 53.1\ ^\circ\text{C}$, against a measured $55.0\ ^\circ\text{C}$, a residual of $1.9\ ^\circ\text{C}$.

Step 5, be critical, which is Criterion D.

One residual is not an evaluation. Plot all the residuals: if they are positive early and negative later, the error is systematic, not random, and the model is structurally wrong rather than merely imprecise. Physically, evaporation removes heat quickly at the start, which the lumped model ignores, and the room does not stay exactly at $22\ ^\circ\text{C}$.

Step 6, extend, which is where the top of Criterion E lives.

Fit $k$ by least squares over all the data rather than from two points, and compare. Then add an evaporative term, $\frac{dT}{dt} = -k(T - T_s) - c$, solve it, and test whether the extra parameter genuinely improves the residuals or just fits the noise. That sequence, derive, fit, test, criticise, extend, is the shape of a 6 out of 6 exploration.

Criteria C and D together are 6 marks, almost a third of the exploration, and they are the two most commonly written as filler.

Personal engagement is evidenced by actions.

Saying "I have always been fascinated by exponential functions" earns nothing. What earns marks is: you collected your own data, you chose between two models and justified the choice, you built a small simulation to test an intuition, you asked a question the source material did not answer, you adapted a standard method because the standard method did not fit your context. Write those actions in the first person and in the past tense at the moment they happen in the argument, not in a separate paragraph at the end.

Reflection must be mathematical and it must appear throughout.

Limited reflection describes: "the model was quite accurate". Meaningful reflection evaluates against evidence: "the mean absolute residual of $1.4\ ^\circ\text{C}$ over ten readings is comparable to my thermometer resolution, so the discrepancy is within measurement uncertainty". Critical reflection changes the direction of the work: "because the residuals are systematically signed, refitting the same model is pointless, so I modified the assumption instead". The strongest explorations have three or four reflective moments spread across the document, each one followed by a decision, and a final reflection that states the limits of what has been shown.

A reliable test on your draft: highlight every sentence that would still be true if you had chosen a completely different topic. Those sentences are worth zero on C and D. If they are the majority of your reflection, rewrite around the specific numbers you actually produced.

The same discipline is what separates a solid Diploma candidate across the whole programme, and it is the habit behind the approach in our IB exam preparation strategy.

The exploration is not written under exam conditions, but it is written under pressure, usually in the same term as the Extended Essay and two other internal assessments. Schools set an internal deadline months before the IB one, and your teacher may give written feedback on one draft only. That single draft is your one piece of examiner style feedback, so it must be a real attempt, not an outline.

A schedule that works in eight weeks:

• Weeks 1 and 2. Two candidate topics, one page of exploratory mathematics on each. Choose. Write the title as a question and the aim in one sentence.
• Week 3. Collect data or generate examples. Do the theory reading. Build the mathematical core on paper, not in a word processor.
• Weeks 4 and 5. Write the full draft, beginning to end, including the extension. Deliberately leave the introduction until last: you cannot introduce an argument you have not made.
• Week 6. Submit the draft for the one round of feedback. While you wait, do the notation pass and build every graph properly.
• Week 7. Act on the feedback. Deepen the mathematics if Criterion E was flagged, since that is where the marks are, and cut anything that does not serve the aim.
• Week 8. Bibliography, figure numbering, final read aloud, then hand in a working week before the deadline so a printer failure is not a crisis.

If you are already behind, the priority order is fixed by the mark allocation. Fix Criterion E first, because it is 6 marks and no amount of polishing rescues mathematics that is below your course level. Then B, because notation is fast to repair. Then D, because two or three genuine reflective paragraphs tied to your numbers can move you a band. Then A, then C. What you must not do is add pages: the fastest gains in the last week almost always come from deleting.

Budget the work properly against the rest of the Diploma. Many students underestimate how much of the year the internal assessments actually consume, a problem we look at across subjects in our piece on the challenges of IB Mathematics.