IB Math IA Examples
All Four Courses

• Volume of revolution: \(V = \pi\displaystyle\int_a^b [f(x)]^2\,dx\)
• Surface area of revolution: \(SA = 2\pi\displaystyle\int_a^b f(x)\sqrt{1 + [f'(x)]^2}\,dx\)
• Improper integrals \(\displaystyle\int_1^{\infty} = \lim_{b\to\infty}\int_1^{b}\) and convergence vs divergence
• Comparison test: if \(g(x) \le f(x)\) and \(\displaystyle\int g\) diverges, then \(\displaystyle\int f\) diverges

Some of the best HL explorations begin with a result that sounds impossible. Gabriel's Horn is one: a shape you can fill with paint but cannot paint. Students meet improper integrals in the HL syllabus and often wonder what they are for. The question can a shape really have finite volume and infinite surface area, and what does that mean? turns an abstract technique into a puzzle worth solving.

Personal engagement in a pure-maths IA does not come from data; it comes from the questions you ask of the result. Why does one integral converge and the other diverge? Does the paradox survive if we change the curve? Is there a family of horns, and where is the boundary between finite and infinite? Those are your own questions, and pursuing them is what Criterion C rewards.

The horn is produced by rotating the curve y = 1/x for x ≥ 1 about the x-axis. The volume comes from the disc method: slice the solid into thin discs of radius y and thickness dx, so each has volume πy² dx, and integrate. The surface area comes from the surface-of-revolution formula 2πy√(1 + (y′)²) dx, which adds the slant width of each thin band.

Both integrals run to infinity, so they are improper: define them as limits as the upper bound b → ∞. The thought process is to compute what you can exactly (the volume) and, where exact computation is impossible (the surface area), to bound the integral by something simpler whose behaviour you know. That is the comparison test, and using it correctly is what makes this HL-level rigour.

Resolve the paradox rather than just stating it. Volume converges because the discs shrink like 1/x², whose integral converges; surface area diverges because the bands shrink only like 1/x, whose integral diverges. The deep point is that convergence depends on how fast a quantity shrinks, and ‘1/x’ is exactly the boundary case — the harmonic series in integral form. Explain that painting the inside would need a coat of paint thinner than any positive thickness, which is why ‘fill it but can't paint it’ is not a real contradiction.

Extend by generalising: rotate y = 1/x^p and find for which values of p the volume converges (p > ½) and the surface area converges (p > 1). Tabulate the three regimes. This is your own investigation, it uses the same techniques with more sophistication, and it shows precisely the kind of rigour the HL Criterion E descriptors ask for.

• Coupled first-order ODEs: \(\dfrac{dx}{dt} = \alpha x - \beta xy,\quad \dfrac{dy}{dt} = \delta xy - \gamma y\)
• Equilibrium analysis: fixed points where \(\dfrac{dx}{dt} = \dfrac{dy}{dt} = 0\); Jacobian \(J\) and eigenvalues \(\lambda\)
• Period of small oscillations \(T = \dfrac{2\pi}{\sqrt{\alpha\gamma}}\) from purely imaginary eigenvalues
• Phase-plane analysis (\(x\) vs \(y\) trajectory) and Euler's method for the full orbit

Nature documentaries show populations of predators and prey rising and falling in waves, apparently forever. The Hudson's Bay Company kept records of lynx and hare pelts for a century, so there is real historical data showing roughly ten-year cycles. The question can two simple equations reproduce this cycle, and do they predict the right period? is a classic HL exploration — but the engagement comes from how far you push it, not from the topic itself.

The mathematical signal is two quantities whose rates of change depend on each other. That is a coupled system of differential equations, and analysing it needs HL tools: equilibrium points, linearisation with the Jacobian matrix, and eigenvalues. Choosing a topic where those tools are necessary, not decorative, is how you make the mathematics commensurate with HL.

Let x be the hare population and y the lynx population (in thousands). Hares breed at a rate proportional to their number (αx) and are eaten at a rate proportional to the number of hare-lynx encounters (βxy). Lynx gain from those encounters (δxy) and die at a constant per-capita rate (γy). Those four effects give the Lotka–Volterra equations.

Assumptions are strong and worth listing: unlimited food for hares, lynx eat only hares, no other predators, no seasons, continuous breeding. The plan: find where the populations would be constant (equilibrium), decide what happens near that point (linearise), and compare the predicted cycle length with the historical data. Choosing parameter values from the data — e.g. α from the hare growth rate in years when lynx were scarce — is what connects the model to reality.

The exploration has produced a genuine prediction from first principles and compared it with history. The discrepancy is where the value lies. The linearisation is only valid for small oscillations; the real cycles are large, and for large orbits the Lotka–Volterra period is longer than 2π/√(αγ) — so part of the gap is the approximation itself. Simulate the full system numerically (Euler or a GDC solver) with realistic starting values and measure the period directly; compare with the linear estimate.

Then challenge the assumptions. Hares do not have unlimited food: replace αx with logistic growth αx(1 − x/K) and show that the equilibrium moves and the eigenvalues acquire a negative real part — the cycles now decay towards a stable point, which is closer to how real ecosystems behave. Explaining what each change does to the eigenvalues, and therefore to the behaviour, is sophisticated mathematics used with purpose: exactly what the HL Criterion E descriptors for 5–6 marks describe.

• Prime numbers and the Fundamental Theorem of Arithmetic; \(n = pq\)
• Euler's totient \(\varphi(n) = (p-1)(q-1)\) and Euler's theorem \(a^{\varphi(n)} \equiv 1 \pmod n\)
• Extended Euclidean algorithm to find the modular inverse \(d \equiv e^{-1} \pmod{\varphi(n)}\)
• Modular exponentiation \(a^{b} \bmod n\) by repeated squaring

Every message you send is encrypted, and the encryption rests on a piece of number theory you can do by hand: it is easy to multiply two primes but very hard to split the product back into them. The question how can arithmetic keep a secret, and why can't a fast computer simply undo it? is a natural HL topic because the syllabus's modular arithmetic, Euler's theorem and the Euclidean algorithm are precisely the tools RSA uses.

Make it yours by actually using it: encrypt a message to a friend with small primes, have them decrypt it, then investigate how long brute-force factoring takes as the primes grow. Timing a factoring program on your own laptop for 4-digit, 8-digit, 12-digit moduli produces your own data and a graph of exponential growth — personal engagement and mathematics together.

RSA works with three numbers: a modulus n that everyone knows, a public exponent e that everyone knows, and a private exponent d that only the receiver knows. Encryption raises the message to the power e modulo n; decryption raises the result to the power d modulo n. The whole system works because e and d are chosen so that the two operations undo each other — and that choice depends on Euler's totient φ(n), which you can only compute if you know the prime factors of n.

The thought process for the exploration: build a working example small enough to verify every step by hand (two-digit primes), then explain why it works using Euler's theorem, then explain why it is secure using the difficulty of factoring. Three layers — do, prove, evaluate — map neatly onto Criteria E, E and D.

Now prove why it works. Euler's theorem says M^φ(n) ≡ 1 (mod n) whenever gcd(M, n) = 1. Since ed = 1 + kφ(n) for some integer k, we get M^ed = M · (M^φ(n))^k ≡ M · 1 = M (mod n). Decryption therefore always returns the original message. Writing this proof in your own words, and handling the edge case where M shares a factor with n, is rigour of the kind the top HL band requires.

Then evaluate security. The attacker knows n and e and needs d, which needs φ(n), which needs p and q. Factoring a 20-digit n is instant for a laptop; a 600-digit n would take longer than the age of the universe with current algorithms. Measure this yourself: time a simple trial-division factoriser on moduli of increasing size and plot time against digits on a log scale — the straight line is exponential growth. Discussing why quantum computers threaten RSA (Shor's algorithm factors in polynomial time) makes a strong, well-informed reflection.

• Fourier series: \(f(x) = \dfrac{a_0}{2} + \displaystyle\sum_{n=1}^{\infty}\big[a_n\cos(nx) + b_n\sin(nx)\big]\)
• Coefficient integrals \(b_n = \dfrac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x)\sin(nx)\,dx\) using orthogonality of trig functions
• Even/odd symmetry to simplify the computation (\(a_n = 0\) for an odd \(f\))
• Gibbs phenomenon at discontinuities (overshoot \(\approx 9\%\)); pointwise vs uniform convergence

If you play an instrument or produce music, you already know that the same note sounds different on a guitar and a piano. The reason is that every real sound is a mixture of pure sine waves at different frequencies, and the mixture is the instrument's ‘fingerprint’. Fourier's claim — that any periodic shape can be built from sines and cosines — is astonishing, and the question how many sine waves does it take to make a square wave, and how good is the approximation? turns it into an investigation.

The mathematical signal is integration of trigonometric products over a period, which is HL calculus, combined with infinite series and convergence — also HL. The topic lets you compute, graph, measure error and discover a genuine subtlety (the Gibbs phenomenon) that most textbooks skip. Discovering something for yourself is personal engagement.

A square wave is the simplest ‘hard’ signal: +1 for half the period, −1 for the other half, with instant jumps. Fourier's theorem writes any periodic function as a constant plus a sum of cos(nx) and sin(nx) terms, each with its own coefficient. The coefficients are found by integrating the function against each sine or cosine — the integral picks out how much of that frequency is present.

The thought process: first use symmetry to halve the work (an odd function has only sine terms), then compute the general coefficient b_n as a formula in n, then build the partial sums and measure how far they are from the true square wave. Plan to graph the 1-, 3-, 5- and 11-term approximations on the same axes — the picture is the heart of the exploration.

The exploration answers its question with a precise picture: a square wave is an infinite sum of odd harmonics with amplitudes 4/(πn), the approximation improves everywhere except at the jumps, and near the jumps there is an unavoidable 9% overshoot. Explain why: the series converges pointwise but not uniformly, because the target function is discontinuous. That distinction — pointwise versus uniform convergence — is sophisticated and entirely appropriate to HL.

Connect back to sound. Record a square-wave tone from a synthesiser and look at its spectrum in free software: you will see peaks at the odd harmonics with heights falling as 1/n, exactly as the mathematics predicts. Compare with a triangle wave (coefficients fall as 1/n², so it converges faster and sounds softer). Using the mathematics to explain why a square wave sounds harsh and a triangle wave sounds mellow is the kind of reflection that joins Criterion D to Criterion C.

• Pearson's \(r\) and Spearman's rank correlation \(r_s\) (compare both)
• Linear regression \(\hat{y} = a + bx\) and log-transformed regression \(\hat{y} = a + b\ln(x)\)
• \(R^2\) comparison between linear and non-linear models; residual patterns
• \(t\)-test for \(\rho = 0\); outlier analysis and leverage

Do richer countries live longer? Almost everyone assumes yes, but how strong is the link, is it the same for poor and rich countries, and what shape does it take? Those three questions are an IA. The data is free (World Bank, UN), covers 190 countries, and includes your own country, which makes it easy to write about with real interest.

The mathematical signal is a relationship between two variables that is probably not a straight line — doubling income from $1,000 to $2,000 surely adds more years than doubling from $50,000 to $100,000. That suspicion is the thought process: it tells you to compare a linear model with a transformed (logarithmic) one, which is exactly the AI SL modelling skill set.

Take GDP per capita in US dollars as x and life expectancy at birth in years as y for a sample of 80 countries chosen across all income levels (say every second country in an alphabetical list, to avoid cherry-picking). Draw the scatter diagram first. It will show a steep rise at low incomes flattening into a plateau — the signature of a logarithmic relationship.

Fit two models with your GDC or spreadsheet: a linear model ŷ = a + bx and a logarithmic model ŷ = a + b ln(x). The second is still linear regression — you simply regress y on ln(x) instead of x — so it is fully within the syllabus. Compare them with R², then test the significance of the correlation. Decide in advance what would convince you one model is better; that is scientific method inside a maths IA.

The exploration's answer: yes, richer countries live longer, but with sharply diminishing returns — the first $5,000 of income is worth more years than the next $50,000. Use the model to make a concrete statement: moving from $2,000 to $4,000 per head predicts about 4 extra years; from $40,000 to $80,000, also 4 years but starting from a much higher base.

Then look at the outliers, because they carry the insight. The USA has the highest income in the sample but a life expectancy several years below the log model's prediction; Qatar is similar. Costa Rica and Japan are well above their predictions. Discuss what the model leaves out — healthcare systems, inequality, diet — and emphasise that correlation is not causation: both income and longevity may be driven by stable government and education. Suggest a multiple-regression extension with healthcare spending as a second predictor. That is critical reflection with a specific, testable next step.

• Chi-squared goodness-of-fit test: \(\chi^2 = \displaystyle\sum\frac{(O - E)^2}{E}\)
• Degrees of freedom \(df = k - 1\) (goodness of fit) or \((r-1)(c-1)\) (independence)
• Critical value \(\chi^2_{\text{crit}}\) vs calculated value; \(p\)-value interpretation
• Expected frequencies under \(H_0\); validity condition \(E \ge 5\)

Football fans believe home teams win more often. Is that true, and if so, by how much? Every Premier League result is public, so you can test the belief with a season of data — 380 matches — and reach a definite statistical answer. If you support a team, the question is already yours; your engagement is in the data you choose to collect and the extensions you think of.

The mathematical signal is categorical data: each match is a home win, a draw or an away win. For comparing observed counts in categories with what we would expect under a hypothesis, the chi-squared goodness-of-fit test is the tool — core AI SL content, and one that students often apply mechanically without understanding. Your job is to understand it.

Formulate the hypotheses precisely. H₀: home advantage does not exist, so each of the three outcomes is equally likely. H₁: the outcomes are not equally likely. The ‘equally likely’ model is a deliberate simplification — you could instead use the away-win proportion as the baseline — and discussing why you chose it is part of the modelling.

The test compares observed counts with expected counts. Under H₀ each outcome is expected 380/3 times. The chi-squared statistic adds up, for each category, (observed − expected)² divided by expected; large values mean the observations are far from the hypothesis. Before using the test, check its condition: every expected count must be at least 5. Here they are 127, so the test is valid — say so.

Quantify the advantage, not just its existence. Home teams won 47% of matches, away teams 34%, so a home team is about 1.4 times as likely to win as an away team. Compute a 95% confidence interval for the home-win proportion (0.47 ± 1.96√(0.47 × 0.53/380) ≈ 0.42 to 0.52) and show that a third lies well outside it.

Then ask the question that turns this into a genuine investigation: why? The 2020–21 season was played without crowds. Repeat the test on that season — the home-win rate fell to about 38% — and run a chi-squared test of independence comparing the two seasons. If the advantage shrank significantly when fans were absent, you have evidence that crowds (not travel or familiarity) drive it. Designing that comparison yourself is the clearest possible demonstration of personal engagement and critical reflection in a statistics IA.

• Geometric series \(\displaystyle\sum_{k=0}^{n-1}(1+r)^k = \frac{(1+r)^n - 1}{r}\) to derive the amortisation formula
• Monthly payment \(M = P\,\dfrac{r(1+r)^n}{(1+r)^n - 1}\)
• Recurrence for the outstanding balance: \(B_{k+1} = B_k(1+r) - M\)
• Sensitivity analysis: how \(M\) changes with \(r\) (\(\dfrac{dM}{dr}\)) and with the term \(n\)

At some point every family talks about buying a home, and the conversation always comes down to one number: the monthly payment. Where does that number come from, and why does a small change in the interest rate make such a big difference? If your parents have a mortgage, or you are curious about a flat you have seen advertised, the question what does this loan really cost? is immediately yours.

The mathematical signal is a sequence of payments with compound interest — geometric sequences and series, the time value of money, and exponential growth, all core AI SL. The depth comes from deriving the payment formula from a geometric series rather than pressing the TVM button, and then using the model to ask ‘what if?’ questions.

A mortgage is a loan of P repaid by n equal monthly payments M, while the outstanding balance grows by the monthly interest rate r_m every month. The thought process: the balance after each month is (previous balance × (1 + r_m)) − M. Writing out the first few months shows a pattern — the terms form a geometric series — and summing that series gives the formula for M that makes the balance exactly zero after n payments.

Assumptions: a fixed interest rate for the whole term, payments made on time, no fees or early repayments. Each is a lever for the reflection section. Use realistic Dubai numbers: a 500,000 AED loan over 25 years at 3.5%, then re-run at 6% to see the effect of a rate rise.

The exploration answers the family question with real numbers and, more importantly, explains why rate changes hurt so much. Build an amortisation table (balance, interest portion, principal portion for each month) and graph the two portions over time: in the early years most of each payment is interest, and the balance barely moves. Explain this from the formula — interest is charged on the whole outstanding balance, which is largest at the start.

For reflection, model the ‘what ifs’ a real borrower faces: a 20-year term instead of 25 (higher payment, far less total interest), an extra 500 AED per month (compute how many months it saves), a variable rate that rises after year 5. Plot M against r for rates from 1% to 10% and comment on the curvature. Then state the limitations honestly — fixed rate, no fees, no inflation (251,500 AED in 2050 is worth less than today) — and suggest adjusting for inflation as the natural extension.

• Logistic growth model: \(P(t) = \dfrac{K}{1 + Ae^{-rt}}\)
• Carrying capacity \(K\) as a horizontal asymptote; inflection at \(P = \dfrac{K}{2}\)
• Parameter fitting: \(A = \dfrac{K - P_0}{P_0}\), then solve for \(r\) from a data point (or linearise \(\ln\frac{K-P}{P} = \ln A - rt\))
• Compare logistic vs exponential vs linear: \(R^2\) on 1960–2022 data and plausibility of forecasts

World population passed 8 billion in 2022. Will it reach 10, 12, 20 billion? A straight-line extrapolation gives a frightening answer, but intuition says growth must slow — the planet is finite. The question what is the right model for population growth, and when will it stop growing? is a modelling IA with a built-in comparison between a naive model and a better one.

The mathematical signal is growth that starts exponential and then saturates. The logistic function is the AI SL model for exactly that: it has a carrying capacity built in. The exploration's thought process is to fit the logistic curve to historical data, then test its prediction against the UN's own forecasts, which were produced by far more complex methods.

The logistic model is P(t) = K / (1 + Ae^−rt). K is the carrying capacity the population approaches, r controls how fast it grows in the early phase, and A is fixed by the starting value. Measure t in years from 1960 (where good data begins) and use UN population figures at ten-year intervals.

Fitting strategy: choose K from the literature (UN and ecological estimates centre on 10–12 billion), fix A using the 1960 population, and solve for r using one recent data point. Then check the model against all the intermediate years you did not use. Explain the choice of K carefully — it is the model's biggest assumption and the obvious thing to vary later.

Interpret the model's shape, not just its numbers. The inflection point (where growth is fastest) is at P = K/2 = 6 billion, which the model places around 1999 — and the real growth rate did indeed peak in the late 1960s–1990s. Compute the year the model reaches 11 billion and the year it reaches 11.9 billion to show how slowly it approaches K.

Then stress-test the biggest assumption: refit with K = 10 and K = 14 billion and tabulate the 2050 and 2100 predictions. The 2050 figures barely move (the data pins the curve), but the 2100 figures diverge widely — an honest statement about what the model can and cannot forecast. Compare with a straight-line fit (which predicts about 11.5 billion for 2050 and keeps rising forever) and with an exponential fit (worse still). Explaining why the logistic model wins — it encodes a physical limit — is the core of your reflection.

• Transition matrix \(P\) (rows = current state, columns = next state), each row summing to 1
• State evolution: \(\mathbf{s}_{n+1} = \mathbf{s}_n P\), so \(\mathbf{s}_n = \mathbf{s}_0 P^n\)
• Steady state: solve \(\boldsymbol{\pi}P = \boldsymbol{\pi}\) with \(\displaystyle\sum_i \pi_i = 1\)
• Eigenvalue view: \(\boldsymbol{\pi}\) is the eigenvector for \(\lambda = 1\); convergence rate set by \(|\lambda_2|\)

Phone brands fight for loyalty, and students have strong opinions about Apple versus Samsung. Suppose you survey your year group now and again in a year: what fraction switched? If the switching pattern stays the same, where does the market end up in the long run — and does it matter where it started? That question — can I predict the long-run market share from this year's switching behaviour? — is an AI HL exploration with your own survey data at its heart.

The mathematical signal is a system that moves between a finite set of states with fixed probabilities, where the next state depends only on the current one. That is a Markov chain, and the tools — transition matrices, matrix powers, steady-state vectors and eigenvalues — are AI HL content that most students meet only in abstract exercises. Applying them to data you collected is what makes it an exploration.

Define three states: Apple, Samsung, Huawei (or ‘other’). The transition matrix P has one row per current brand and one column per next brand; entry (i, j) is the probability that a user of brand i is using brand j a year later. Each row sums to 1 because every user ends up somewhere. Build P from your survey: of the Apple users at the start, what fraction were still on Apple a year later, and so on.

The state vector s is a row of market shares. One year on, the shares are sP; two years on, sP²; after n years, sPⁿ. The steady state π is the share vector that no longer changes: πP = π. Assumptions to state: transition probabilities are constant over time, each switch depends only on the current brand (not the full history), and your year group represents the market. All three are questionable and all three give you reflection material.

The exploration's answer is a definite prediction — Apple settles at about 48% whatever the starting shares — and a timescale: roughly a decade. Test it: start from s₀ = (0.2, 0.2, 0.6) and show the chain converges to the same π. That independence from the start is the key property of a regular Markov chain, and demonstrating it with your own matrix is both rigorous and engaging.

Reflect on the assumptions with evidence. Transition probabilities are not constant — a new model launch changes them — so the chain is really time-dependent; discuss what happens if P changes every few years. The ‘memoryless’ assumption fails if people who switched once are more likely to switch again; propose a second-order chain with six states (brand now, brand before) as an extension. And your sample is one year group in one school — say what a national survey might change. Each of these is specific, mathematically framed and testable: that is what ‘critical reflection’ means.

• Weighted directed graph \(G = (V, E)\): vertices (stations), edge weights (journey time)
• Dijkstra's algorithm: greedy relaxation \(d(v) = \min\big(d(v),\ d(u) + w(u,v)\big)\); \(O(V^2)\) complexity
• Adjacency matrix representation and path reconstruction from predecessor labels
• Sensitivity: recompute with \(\times 1.5\) weight on congested edges; threshold at which the optimal path changes

Every day you take the fastest route somewhere, and your phone computes it in a fraction of a second. How? The Dubai Metro map, with its interchanges and different journey times, is a graph — and finding the quickest route between two stations is the shortest-path problem. The question what is the fastest route from my station to the airport, and how does congestion change it? is concrete, local and lets you collect your own timings.

The mathematical signal is a network with weighted edges and a question about an optimal path. That is graph theory, and the specific tool is Dijkstra's algorithm — AI HL content. An algorithm is only understood when you can run it by hand and explain why it is guaranteed to work, so the exploration should do both.

Model stations as vertices and direct connections as edges weighted by travel time in minutes (measured on real journeys, or from the official timetable). Keep the network small enough to run the algorithm by hand — five or six key stations — and say why you simplified. The thought process behind Dijkstra: always extend the shortest known route first, because any route that leaves it later cannot be shorter.

Plan two phases. First, find the shortest route in normal conditions and verify it by listing every alternative. Second, change edge weights to reflect rush-hour congestion and see whether the optimal route changes — that ‘what if’ analysis is where the exploration gets its depth.

The exploration answers its question — 10 minutes via Union, ADCB and Deira — and shows the answer is robust to congestion on one link. Push further: find the weight at which A→B must rise before the direct A→C route becomes optimal (solve 3 + 2 > 6 ⇒ AB > 4). That threshold analysis turns a single answer into an understanding of the network's sensitivity.

Explain why Dijkstra is correct: when a vertex is chosen as the smallest unvisited, every other route to it must pass through some unvisited vertex whose label is already larger, so it cannot be shorter — provided no edge has negative weight. State that condition and note that travel times are always positive, so the algorithm applies. Compare its efficiency with checking every path (there are only a few here, but the number of paths grows exponentially with network size, which is why real apps use Dijkstra or its descendants). Suggesting an extension to the full 50-station network using a spreadsheet implementation, or to time-dependent weights, shows you understand the algorithm's scope.

• Linear objective function \(P = c_1 x + c_2 y\) with decision variables \(x, y\)
• System of linear inequality constraints \(a_i x + b_i y \le k_i\) defining the feasible region
• Vertex enumeration: evaluate \(P\) at each corner point (optimum of a linear function lies at a vertex)
• Simplex method as the algorithmic extension for three or more variables

A small business with limited ingredients, limited staff hours and limited oven time has to decide every morning what to bake. More croissants or more loaves? The owner's instinct is good but not optimal. The question what production mix maximises daily profit under these limits? is the classic linear programming problem — and if a family member runs any small business, you can use their real numbers.

The mathematical signal is an objective (profit) to be maximised subject to several linear constraints (resources). The AI HL tool is linear programming: express the constraints as inequalities, draw the feasible region, and use the fact that the optimum of a linear objective over a convex polygon is always at a vertex. Understanding why it is at a vertex is what separates a strong exploration from a mechanical one.

Define decision variables: x croissants and y loaves per day. Each uses known amounts of flour, labour and oven capacity; each earns a known profit. Every resource limit becomes a linear inequality, and the profit becomes a linear objective function. Non-negativity constraints (you cannot bake a negative loaf) complete the model.

Assumptions: profit per unit is constant (no bulk discounts), resources can be divided continuously (fractions of loaves are allowed — discuss rounding), and everything baked is sold. The thought process: sketch the constraints, identify the feasible polygon, find its corners exactly by solving pairs of equations, evaluate the profit at each corner, and pick the best. Then check the winner satisfies every constraint — a step that catches errors examiners love to find.

The exploration answers the business question and can go on to answer the ones the owner will ask next. Sensitivity analysis: if croissant profit rises to 10 AED, does the optimal corner change? (Re-evaluate; (180, 0) now gives 1800 and wins — the plan flips entirely.) Shadow prices: if one more kilogram of flour were available, how much extra profit would it bring? Re-solve with 31 kg and compute the difference. These are exactly the questions linear programming was invented to answer, and asking them shows mathematical maturity.

Reflect on the model's limits: demand is not unlimited (nobody sells 180 croissants every day), integer solutions may be needed (discuss what happens when the optimal vertex is not a whole number), and profits are not perfectly linear. Suggest adding a demand constraint x ≤ 150 and re-solving, or extending to three products, which can no longer be drawn and motivates the simplex method. Mentioning the simplex method as the next step — and why the graphical method cannot scale — is a well-informed reflection.

• Linear regression \(\hat{y} = a + bt\) with \(R^2\) comparison
• Logistic model \(P(t) = \dfrac{K}{1 + Ae^{-rt}}\) fitted by linearisation \(\ln\dfrac{K-P}{P} = \ln A - rt\) or non-linear least squares
• Residual analysis to evaluate model fit (not just \(R^2\))
• Forecast comparison and the physical bound \(0 \le P \le 100\%\)

Norway went from almost no electric cars to over 80% of new sales in little more than a decade. Looking at the data, the temptation is to draw a straight line and read off the future — and the straight line says more than 100% of cars will be electric by 2030, which is absurd. That absurdity is the IA. The question which model correctly describes technology adoption, and what does it predict? teaches the most important lesson in modelling: a good fit is not the same as a good model.

The mathematical signal is a quantity that must stay between 0% and 100% and that grows slowly, then fast, then slowly again. That is a logistic curve, and the exploration's thought process is a head-to-head comparison: fit a linear model and a logistic model to the same data, compare their fit statistics and their predictions, and explain why the one with the physical ceiling wins.

Let t be years since 2010 and P the percentage of new cars sold that are electric. The linear model is P = a + bt. The logistic model is P = K/(1 + Ae^−rt) with K fixed at 100% because the share cannot exceed that; A is fixed by the 2010 value and r is fitted by least squares (or by linearising: ln((K − P)/P) = ln A − rt is a straight line in t, so r is a gradient).

Decide the comparison criteria before fitting: R² on the historical data, the pattern of residuals, and the plausibility of the 2030 prediction. Including plausibility as a criterion is the point of the exploration — statistics alone cannot tell you a model is wrong if its predictions are impossible.

The exploration's conclusion is a principle, not just a number: model choice must be driven by the structure of the situation (a bounded percentage) as well as by fit statistics. Explain why the linear model fitted so well despite being wrong — over the middle of an S-curve, where most of the data lies, a logistic curve is nearly straight — and why that makes extrapolation dangerous.

Then reflect on the logistic model's own assumptions. K = 100% assumes every buyer eventually switches; if 5% of buyers never will (rural users, specialist vehicles), K should be 95% and the forecasts change — refit and show the effect. The constant r assumes no policy shocks; Norway's incentives changed in 2023, so test whether a fit to 2010–2022 predicts 2023 well. Compare with another country at an earlier stage (e.g. the UAE) and use the Norwegian r to forecast it, discussing why the rate may differ. Each of these is a specific, mathematically framed question, and that is what critical reflection looks like in a modelling IA.

Coherent = it flows, each part leads to the next. Organised = clear sections, headings, page numbers. Concise = nothing wasted — no padding, no repeated calculations, no pages of raw data in the middle.

• Title states the actual question — “Modelling the cooling of tea using Newton’s law”, not “Maths IA”.
• Introduction has three things: the aim, the rationale (why you care) and a short plan of the maths.
• Clear headed sections: Introduction → Data/Method → Mathematical work → Results → Evaluation → Conclusion → Bibliography → Appendix.
• Page numbers on every page; total 12–20 pages (bibliography and appendix not counted).
• Every graph, table and figure is numbered and referred to in the text (“as shown in Figure 3”).
• Flow sentences between sections: “Now that k has been found, the model can be tested against the remaining data.”
• Repetitive calculations go in the appendix — one full worked example in the body, then a summary table.
• Raw data lives in the appendix; the body has a clean summary table.
• Conclusion answers the aim in its first sentence, using the same words as the introduction.
• Read aloud once, then give it to someone who does not do maths and ask where they got lost.

• Going over 20 pages — “concise” is lost, so the maximum becomes 3/4.
• Writing it like a diary: “First I tried this, then I tried that…”
• A two-page history of the topic or the mathematician’s biography.
• Graphs with no axis labels, no units, no title, never mentioned in the text.
• A conclusion that introduces new maths or new ideas.

• Define every variable the first time it appears, with units: “Let T be the temperature in °C and t the time in minutes.”
• Same symbol for the same thing everywhere — if k is the cooling constant on page 3 it must not become c on page 7.
• Correct notation: dT/dt as a proper derivative; ≈ when rounding; = only when truly equal.
• Use an equation editor (Word, Google Docs, LaTeX). Never type x^2 or sqrt(x).
• Correct terminology: “gradient”, “the model predicts”, “statistically significant at the 5% level”.
• Every graph: title, labelled axes with units, sensible scale, a key if more than one line.
• Every table: column headings with units, consistent decimal places.
• Consistent rounding — decide (e.g. 3 s.f. for final answers) and state it once.
• Use several forms — words, algebra, a graph, a table — each adding something new.
• Number equations you refer back to: “substituting (2) into (1) gives…”
• Degrees vs radians stated; domain of every function stated (“for t ≥ 0”).
• Calculator or spreadsheet output is translated — write underneath what the numbers mean.

• Calculator notation in the text: 2E-3, x^2, * for multiply.
• Switching between y, f(x) and T(t) for the same function.
• Desmos/GeoGebra graphs pasted with the default “x” and “y” axis labels.
• “Proof by GDC”: “my calculator gave r = 0.812” with no formula shown.
• Writing = in a chain where you actually rounded.

The secret: engagement is shown, not told. “I am passionate about football” earns nothing. Collecting 300 real match results yourself, setting your own hypothesis and testing it — that earns marks.

• A question that is genuinely yours, linked to your life — your sport, hobby, family business, your city.
• Primary data where possible — you measured the tea, timed the metro, surveyed 60 students. Say exactly how.
• Your own decisions, explained: “I chose a logistic model rather than exponential because population cannot exceed the carrying capacity.”
• Something beyond the textbook version — derive a formula instead of quoting it; build the Markov matrix from your own observations.
• Independent thinking — test an assumption, compare two models, ask “what if the rate were 6% instead of 3.5%?”
• Creativity in approach — an unusual application, or a familiar method applied to something new.
• Written in your own voice: “I expected home advantage to be small — the result surprised me.”
• Visible in every section, not just one paragraph in the introduction.

• Topics copied from popular lists (“golden ratio in art”) with nothing new added — examiners have read thousands.
• A downloaded data set with no explanation of why it matters to you and no manipulation of it.
• Over-claiming: “I have been fascinated by differential equations since I was five.”
• Letting a teacher, tutor or AI tool do the thinking — you sign a declaration that the work is your own.

• Reflect throughout, not only in the conclusion — 2–4 sentences of “so what?” after each result.
• Check results against reality — find an outside number to compare with (the UN population figure, a published rate).
• State every assumption and say what happens if it is false.
• Discuss limitations with their effect: sample size, measurement error, model simplification, time period.
• Consider an alternative method and compare (linear vs logistic, Euler vs exact solution).
• Suggest specific extensions — not “more research could be done” but “repeat with a lidded cup to test whether evaporation is the main heat-loss route”.
• Link back to the aim — did the maths answer the question fully, partly, or not at all?
• Reflect on the maths itself — why this technique was suitable, where it struggled.
• Be honest about surprises and failures: “My first model gave a negative height, which showed I had mis-substituted the volume constraint.”

• A single “Limitations” bullet list at the end with no effect on the results explained.
• Generic sentences that fit any IA: “human error may have affected results”.
• Reflection that is only emotional (“I enjoyed this”) and not mathematical.
• Calling a model “accurate” without a number — percentage error, residuals, R² and a reality check.

Commensurate = at the level of your course. Correct = no errors anywhere. Precise = correct and careful: exact values kept, rounding done late, domains and units carried through. Sophistication (HL) = using the maths in a thoughtful, non-routine way — comparing approaches, deriving rather than quoting. Rigour (HL) = justifying every step — checking conditions, proving convergence, verifying the second derivative.

• The maths is needed to answer the question — if removing a section leaves the aim answered, cut it.
• At least one piece of maths is at or beyond your course level — build the IA around it.
• Show the method, not just the answer: formula → substitution → simplification → result. Technology checks; it does not replace working.
• Every formula is derived (best, especially HL) or quoted with a source and the conditions under which it applies.
• Conditions are checked before a method is used — expected frequencies ≥ 5 for χ²; gcd(e, φ(n)) = 1 for RSA; second-derivative sign for a minimum.
• Each step explained in one sentence: “Taking logs turns the exponential into a straight line, so k can be read as the gradient.”
• Accuracy managed — exact values kept as long as possible; round only at the end, to a number of s.f. that matches the data.
• Results verified a second way — substitute back; compare Euler with the exact solution; decrypt the RSA message.
• Errors quantified — percentage error, residual plot, or comparison with a known value.
• (HL) Sophistication — compare two methods, generalise a result, derive the formula you use, discuss sensitivity to parameters.
• (HL) Rigour — no unexplained jumps; limits, domains and special cases handled (what if r = 0 in the mortgage formula?).
• Zero mathematical errors — check every line twice, then have someone else check it.

• Twelve pages of simple statistics in an HL IA — relevant but not commensurate, stuck at 3/6.
• A very advanced technique used badly, with errors and no understanding — 3–4/6.
• “The calculator gave k = 0.04463” with no formula and no substitution.
• A formula from Wikipedia with no statement of when it is valid.
• A hypothesis test with no H₀ and H₁, no significance level, or no conclusion in context.
• Confusing correlation with causation in the interpretation.

• Title states the question.
• Aim, rationale and plan in the introduction.
• Clear headed sections in a logical order.
• Page numbers; 12–20 pages.
• All figures and tables numbered and referred to in the text.
• Raw data and repetition in the appendix.
• Conclusion answers the aim directly.
• A non-maths reader could follow it.

• Every variable defined with units at first use.
• Same symbols throughout.
• Equation editor used; no calculator notation.
• Graph axes labelled with units; titles; keys.
• Consistent rounding, stated once.
• Correct use of =, ≈, ⇒, ∴.

• Topic clearly linked to your own life or interest.
• Own data or own model-building, explained.
• Own decisions justified (why this model, why this test).
• Something beyond the textbook version.
• Engagement visible in every section.

• Short reflection after every main result.
• Results compared with a real-world value.
• Each assumption stated with the effect if it fails.
• Limitations explained with their effect on the answer.
• An alternative method considered and compared; specific extensions suggested.

• Core maths at (or above) your course level.
• Every step shown: formula → substitution → result.
• Formulas derived or justified with conditions; conditions for each method checked.
• Every result verified a second way.
• Zero errors — checked twice by you, once by someone else.
• (HL) At least one place showing sophistication and rigour.