Photon Students
For Photon Academy students, parents and tutors · iPhone and iPad
Open in the App Store
IB Diploma
Before the Diploma
Exam preparation
Maths IA · AA and AI · SL and HL
Here is our full list of Maths IA topics, in 12 families. Each one has a title, a one-line hook, how well it fits AA and AI at SL and HL, how demanding it is and the syllabus topics it uses. Our IB Maths teachers at Photon Academy made the list.
Choosing the topic is the hardest part of the Maths IA. This list is short on purpose: enough to pick from, not the full plan. Each topic has a free guide, and Photon Academy students get help to make it their own.
The Maths IA is an individual written exploration, marked out of 20 and worth 20% of your final grade. It is the same task at SL and HL, in AA and AI. The report is about 12 to 20 pages, and the IB suggests about 10 to 15 hours of teaching time. So the topic has to be one you can finish well in that time.
Got an idea already? Get its research question checked, free.
Free research question checkThe first RQ Check is free · scored on 5 points
First RQ Check free · scored on 5 points
Most topics suit both AA and AI, and every family has several to choose from. The chart at the top shows the families. These show the area of maths, the difficulty and what the fit dots mean.
Showing 97 of 97 ideas
Loans, amortisation, credit cards, savings, retirement and the real cost of borrowing.
Owe 3,000 on a card at 26% a year, pay only the minimum, and the debt takes nearly 18 years to clear. You're going to find out why it takes so long, and what a slightly bigger payment does to it.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AA 1.3, 1.4, 1.7, 2.9, 5.18 HLAI 1.3, 1.4, 1.7, 1.9 HL, 2.9 HL, 5.14 HL
A lender says 3% a year, flat, for 5 years. The real rate on what you actually owe is closer to 5.6%. You're going to find why the two rates differ, and how to turn one into the other.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AA 1.3, 1.4, 2.10, 1.7AI 1.3, 1.4, 1.6, 1.7, 1.8
Pay the bank the same amount every month for 25 years, and in year one most of it never touches the loan. You're going to find the month that flips, and what a rate rise does to it.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AA 1.3, 1.4, 1.7, 2.9, 5.6, 5.7AI 1.3, 1.4, 1.7, 1.9 HL, 5.9 HL
Two loan offers, same amount, different small print. Which is cheaper, and when does the answer flip? You're going to prove it with algebra, not just a calculator.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleModelling
AA 1.3, 1.4, 1.7, 1.8AI 1.3, 1.4, 1.5, 1.7, 1.8
Buying costs a deposit and fees up front, and then more each year than renting. Yet in the model below it overtakes renting after about 8 years. You're going to find the break-even year, and what moves it.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingModelling
AA 1.3, 1.4, 1.8, 2.9, 2.10, 5.11AI 1.3, 1.4, 1.7, 1.8, 5.5
A fund of 1,000,000 earning 5% a year. Withdraw 50,000 a year and it never runs out. Withdraw 60,000 and it's gone in under 37 years. You're going to find the line between the two, and what inflation does to it.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AA 1.3, 1.4, 1.7, 1.8, 2.9, 4.9AI 1.3, 1.4, 1.7, 1.11 HL, 4.9
Divide 72 by the interest rate and you get the years to double your money. It's off by under 1% at 8%, and by about 13% at 40%. You're going to find where it works and why 72 was chosen.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumAlgebra and functions
AA 1.4, 1.5, 1.7, 2.9, 5.13 HL, 5.19 HLAI 1.4, 1.5, 1.6, 1.9 HL, 2.3
How banks create money, how investors spread risk, how options are priced and how inequality is measured.
Same budget, several ways to spend it. The best plan sits on a corner, and a small price rise can send it to another corner. Your job is to see it coming.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AA 1.16 HL, 2.1AI 1.8, 2.1, 2.5, 2.6, 1.14 HL
A bank cannot know who will repay. It can only estimate a probability of default and choose where to draw the line. You're going to build that estimate, find the best line, and ask who it shuts out.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingModelling
AA 2.9, 4.7, 5.8AI 2.9 HL, 2.10 HL, 4.7, 4.13 HL, 5.9 HL
You have a bet where you win more often than you lose. How much of your money should you put on it? Too little wastes the edge, and too much can ruin you even though the odds are in your favour.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumCalculus
AA 1.7, 2.9, 4.7, 4.8, 5.6, 5.8AI 1.5, 1.9 HL, 4.7, 4.8, 5.6, 5.9 HL
Two countries can have the same average income and very different lives. The Gini coefficient squeezes that gap into one number. You're going to see where the number comes from, and what it hides.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumCalculus
AA 2.9, 5.3, 5.5, 5.11AI 2.5, 2.10 HL, 4.4, 5.5, 5.8
Someone deposits 1,000. The bank lends most of it out, and that loan is deposited at another bank, which lends again. You're going to find how far this chain can go, and whether banks really work that way.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AA 1.3, 1.7, 1.8, 2.9, 5.6AI 1.3, 1.9 HL, 1.11 HL, 2.5, 2.9 HL
An option is a bet on a share price, and yet its fair price does not depend on how likely you think the price is to rise. You're going to find out why that is true, and build the price from coin flips.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingStatistics and probability
AA 1.4, 1.9, 2.9, 4.7, 4.8, 4.9AI 1.4, 4.7, 4.8, 4.9, 4.15 HL
Raise the price and you sell less. Lower it and you earn less per sale. Somewhere between is the best price, and you're going to find it from real data.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AA 2.1, 2.9, 4.4, 5.3, 5.7, 5.8AI 2.5, 4.4, 5.6, 5.7, 5.10 HL, 4.13 HL
Mix a risky share with a calm one and the mixture can be safer than either. You're going to find the exact mix with the least risk, and see what decides how much you gain.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumStatistics and probability
AA 4.3, 4.4, 4.7, 2.6, 5.8, 4.14 HLAI 4.3, 4.4, 4.7, 5.7, 4.14 HL
The maths inside Bitcoin, wallets, exchanges and encrypted messages: series, probability, modular arithmetic and curves.
Some exchanges have no buyers and sellers at all. A pool holds two kinds of token, and one rule sets every price: the amounts multiplied together never change. You will find out what that rule does to the price you get.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumCalculus
AA 2.8, 5.3, 5.6, 5.7, 5.8, 5.14 HLAI 2.4, 5.3, 5.4, 5.9 HL, 5.10 HL
Every four years or so, the reward for adding a block to Bitcoin's chain is cut in half. That sounds like it should never end, yet the total can never pass 21 million. Your job is to explain why an endless process has a finite total.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumAlgebra and functions
AA 1.3, 1.5, 1.7, 1.8, 2.9AI 1.3, 1.6, 1.11 HL, 2.10 HL
A shop sees your payment but waits for a few more blocks before handing over the goods. Wait for too few and a cheat with enough computing power can reverse it. Wait for too many and nobody buys anything. You are going to find the number that balances the two.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingStatistics and probability
AA 1.3, 1.8, 1.15 HL, 4.5, 4.8, 2.9AI 1.3, 4.5, 4.8, 4.17 HL, 2.10 HL
Take two points on a curve, draw a line, find where it hits the curve again and flip it. That strange recipe is a way to add points, and it is the basis of the keys in many crypto wallets. You will check why the recipe never leaves the curve.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingAlgebra and functions
AA 2.1, 2.3, 2.12 HL, 5.4, 5.14 HLAI 2.1, 5.4, 5.9 HL
In a room of just 23 people, the chance that two share a birthday is already about one half. The same arithmetic decides how long a digital fingerprint must be. Your job is to explain why the number is so small and then use it.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumStatistics and probability
AA 1.7, 2.9, 4.5, 5.19 HLAI 1.6, 4.5, 4.6, 4.12 HL
You can publish a number on the open internet that lets anyone lock a message to you, and still be the only person who can unlock it. Why can't they just work it out?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingAlgebra and functions
AA 1.6, 1.7, 1.9, 1.15 HL, 2.9AI 1.5, 1.6
Routes, rounds, networks, rankings and matchings. Mostly AI HL content; AA students treat it as an extension.
A newspaper round, a gritting lorry and a street sweeper all have to drive every street, and nobody wants to repeat more than they must. You're going to find the fewest wasted kilometres and prove you cannot do better.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AI 3.14 HL, 3.15 HL, 3.16 HL, 1.14 HL, 4.3
In 1736 the people of Königsberg wondered whether anyone could cross all seven of the city's bridges exactly once. Euler showed why not with one idea: count how many roads meet at each place.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AI 3.14 HL, 3.15 HL, 3.16 HL, 4.12 HLAA 1.15 HL, 1.6
Two subjects cannot sit in the same exam session if a single student takes both. How few sessions can a whole school get away with? You're going to find a lower bound, an upper bound, and close the gap.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingModelling
AI 3.14 HL, 3.15 HL, 4.1, 4.2, 4.12 HLAA 1.15 HL
To join four towns with the least cable, you might lay a road between neighbours. Add a junction in the middle and the total gets shorter, provided the roads meet at exactly the right angle.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingGeometry and trigonometry
AI 3.14 HL, 3.16 HL, 3.10 HL, 5.9 HLAA 3.12 HL, 3.13 HL, 5.14 HL, 5.8
A page is important if important pages link to it. That sounds circular, and it is, until you write it as one equation and let a matrix settle the argument. You're going to rank a small web, and test what the ranking depends on.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingModelling
AI 1.14 HL, 1.15 HL, 3.14 HL, 3.15 HL, 4.19 HL, 1.8
Deliver to six places and visit each once: how many routes are there, and does the obvious one win? You're going to find out how far off the quick method can be.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumGeometry and trigonometry
AI 3.14 HL, 3.15 HL, 3.16 HL, 1.6, 4.3
Every student ranks the schools and every school ranks the students. Can you pair them so that no student and school would both rather swap? Gale and Shapley proved you always can, and gave a method that finds it.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingModelling
AA 1.15 HL, 1.6, 4.7AI 3.14 HL, 4.5, 4.7
Best shapes, fastest paths, areas and volumes: classic calculus with a real object to test it on.
Two methods, one area. Which is closer, by how much, and what does it cost? The evaluation practically writes itself.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleCalculus
AA 5.5, 5.11, 2.3AI 1.6, 5.5, 5.8, 2.5
The straight line is the shortest way down a slide. It is not the fastest. You will build the time of a descent as an integral and find out how much time a curve can save.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingCalculus
AA 3.1, 5.3, 5.8, 5.11, 5.15 HL, 5.16 HLAI 5.5, 5.8, 5.9 HL, 5.13 HL
Open the aperture wide and the lens blurs the picture. Close it down and diffraction blurs it instead. So there is a best f-number in the middle, and you can find it with calculus and then check it with a camera.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumCalculus
AA 1.3, 2.7, 2.9, 5.6, 5.8AI 1.3, 2.5, 5.6, 5.7, 2.10 HL, 4.13 HL
Tried by our studentsFull guide
Differentiate once and the textbook 'best can' falls out in a line. Your IA starts where that line stops matching the cans on the shelf.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleCalculus
AA 3.1, 5.3, 5.7, 5.8, 5.14 HLAI 3.1, 5.6, 5.7, 5.10 HL
A hanging chain looks like a parabola, and Galileo thought it was one. It is not. You will measure how big the difference really is, and when it matters.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumCalculus
AA 2.9, 2.3, 5.6, 5.17 HL, 5.19 HL, 5.13 HLAI 2.5, 5.8, 5.9 HL, 2.9 HL
You can run faster than you can swim, so the straight line to the swimmer is not the quickest way. Somewhere along the shore there is a best place to enter the water, and the maths for it also describes how light bends.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumCalculus
AA 3.2, 5.3, 5.6, 5.8, 3.8AI 3.2, 5.3, 5.6, 5.7, 5.9 HL
Real data is jagged, and you cannot differentiate a jagged line. So how do you say when it was changing fastest?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleCalculus
AA 2.9, 5.1, 5.2, 5.6, 5.7AI 2.5, 5.1, 5.2, 5.6, 5.10 HL, 2.9 HL, 4.13 HL
You can pour water into a vase and read its volume. Can you get the same number from a pencil, a ruler and an integral?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumCalculus
AA 2.3, 3.1, 5.11, 5.17 HLAI 1.6, 3.1, 5.5, 5.8, 5.12 HL
Sit too close to a screen and it fills your view badly, sit too far and it shrinks. Somewhere between, there is a best seat, and you can find it exactly.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumCalculus
AA 3.2, 5.3, 5.6, 5.8, 3.9 HL, 5.15 HLAI 3.2, 5.3, 5.7, 5.9 HL, 5.10 HL
Tried by our studentsFull guide
Every glass holds the same drink, but they do not all use the same amount of glass. You are going to look for the bowl shape that needs the least glass for a given volume, and see how far real glasses are from it.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingCalculus
AA 3.1, 5.3, 5.8, 5.11, 5.17 HLAI 3.1, 5.7, 5.8, 5.12 HL
Tried by our studentsFull guide
Every topic above links to a free guide on photonresources.com. Each guide shows three things.
What the topic is about and what you would explore.
The course topics it uses, for AA and for AI.
A route through the exploration, so you can see how the pieces fit.
The guide is where a topic starts, not where it ends. Photon Academy students also get help to make it their own: an aim, a twist and feedback on the draft. Anyone can get a research question checked free with the IA & EE Coach. We give feedback, questions and teaching. We never write or rewrite your IA.
Models of things that change over time: cooling, draining, epidemics, populations, medicine and chaos.
Everyone says a hot drink cools exponentially. You're going to test that with real readings, and measure how wrong it is.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleModelling
AA 1.7, 2.9, 4.4, 5.18 HLAI 2.5, 2.6, 4.4, 4.13 HL, 5.14 HL
Take a tablet every eight hours and the level in your blood does not just repeat. It climbs, then settles. You're going to find the level it settles at, and the longest gap between doses that is still safe and effective.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AA 1.3, 1.5, 1.7, 1.8, 2.9, 5.18 HLAI 1.3, 1.5, 1.11 HL, 2.9 HL, 5.14 HL
One model tracks the whole population with three equations. The other follows every person by the rules of chance. Which one better matches a real outbreak, and where do they disagree?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingModelling
AA 2.9, 2.10, 5.6, 5.18 HLAI 2.6, 4.8, 5.14 HL, 5.16 HL, 5.9 HL
Tried by our studentsFull guide
Start two simulated weather systems almost exactly the same. Wait. They stop looking alike. Can you measure how fast they drift apart, and how far ahead any forecast can be trusted?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingModelling
AA 1.5, 2.9, 4.4, 5.18 HLAI 2.10 HL, 4.4, 5.1, 5.16 HL
Tried by our studentsFull guide
When hares are plentiful, lynx thrive. When lynx thrive, hares crash. Then the lynx starve and the hares come back. Can two equations produce that endless chase, and what does the maths say about the point where nothing changes?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingModelling
AA 5.1, 5.6, 5.18 HLAI 2.6, 5.1, 5.9 HL, 5.14 HL, 5.16 HL, 5.17 HL
A chaser always aims at a runner who keeps moving. The path it traces is not a straight line, and not a circle. Does it ever get there?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingCalculus
AA 3.2, 5.6, 5.9, 5.10, 5.18 HLAI 3.2, 3.12 HL, 5.9 HL, 5.16 HL
A bone is a few thousand years old, or a few tens of thousands. The only clock is a tiny amount of carbon that fades by half every 5,730 years. How accurate is that clock, and where does it stop working?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AA 1.5, 1.7, 2.9, 5.1, 5.6, 5.18 HLAI 1.5, 1.6, 2.9 HL, 5.1, 5.14 HL
A rumour crosses a school in days; a disease takes weeks. Same maths, different numbers. You're going to build the spread yourself and see what really matters.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingModelling
AA 1.11 HL, 2.9, 5.18 HLAI 2.9 HL, 4.5, 5.14 HL, 5.15 HL, 5.16 HL
Water leaves a tank fast at first and slowly at the end. Can a differential equation tell you exactly when it runs dry?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingCalculus
AA 2.6, 5.10, 5.17 HL, 5.18 HLAI 2.5, 5.12 HL, 5.14 HL, 5.16 HL
Chance, risk, sampling and evidence: expected value, Bayes, distributions and hypothesis tests.
In many real data sets, about 30% of numbers start with 1 and fewer than 5% start with 9. Does your data play along, and what happens when it doesn't?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumStatistics and probability
AA 1.5, 1.7, 2.9, 4.5AI 1.5, 4.2, 4.5, 4.11
How do ecologists count fish they cannot see all at once? Catch some, mark them, release them, then catch again and see what fraction are already marked.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumStatistics and probability
AA 4.1, 4.5, 4.7, 4.8, 1.10 HLAI 4.1, 4.5, 4.7, 4.8, 4.14 HL, 4.16 HL
Every game has a rule that decides whether the house or the player wins in the long run. You're going to find it and then test it by playing.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleStatistics and probability
AA 4.5, 4.6, 4.7, 4.8, 4.11AI 4.5, 4.6, 4.7, 4.8
A 6-from-49 lottery has 13,983,816 possible tickets, so one ticket has a chance of about 1 in 14 million. Is there a jackpot big enough that the average ticket stops losing money?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumStatistics and probability
AA 1.9, 4.5, 4.7, 4.8, 4.14 HLAI 4.5, 4.7, 4.8, 4.14 HL, 4.17 HL
Will it rain tomorrow if it rained today? Where will you be on the board after 100 turns? Matrices can say, if you're careful about what they assume.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingStatistics and probability
AI 4.19 HL, 1.14 HL, 1.15 HL, 3.15 HL, 4.5, 4.6
Football goals and rugby tries seem to arrive without warning. If they really do, a Poisson model should predict how many 0–0 draws and 4–3 thrillers a season holds. Does it?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumStatistics and probability
AA 4.5, 4.7, 4.8AI 4.3, 4.5, 4.7, 4.8, 4.11, 4.14 HL, 4.17 HL
Tried by our studentsFull guide
A poll of 1,000 people claims to speak for millions. You're going to find out exactly how much confidence that deserves, and where it runs out.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumStatistics and probability
AA 4.1, 4.8, 4.9, 4.14 HLAI 4.1, 4.8, 4.9, 4.14 HL, 4.15 HL, 4.16 HL
A test that is right 90% of the time says you have a rare condition. Are you probably ill? Your first instinct will be wrong, and showing why is the whole IA.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleStatistics and probability
AA 4.5, 4.6, 4.11, 4.13 HLAI 4.5, 4.6
You will see 100 candidates, one at a time, and must accept or reject each on the spot. There is a rule that picks the single best one more than a third of the time, and its cut-off is about 37% of the way through.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingStatistics and probability
AA 4.5, 4.6, 4.11, 5.6, 5.8, 5.11AI 4.5, 4.6, 5.5, 5.9 HL, 5.10 HL
Tried by our studentsFull guide
You stand in a queue and wonder why it's never short. You're going to count it, model it, and find out why a busy till is worse than it looks.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingModelling
AI 1.3, 1.11 HL, 4.7, 4.8, 4.11, 4.17 HL, 4.19 HL
In a famous set of university admission figures, men were admitted at a clearly higher rate than women overall, yet in most departments women did as well or better. Both statements are true.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumStatistics and probability
AA 4.5, 4.6, 4.11, 4.13 HLAI 4.1, 4.5, 4.6, 4.11
Everyone says "get more sleep". You're going to ask how strong the link to marks really is, and how sure you can be that it isn't just chance.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleStatistics and probability
AA 4.1, 4.3, 4.4AI 4.1, 4.3, 4.4, 4.10, 4.11, 4.12 HL, 4.13 HL
You've waited for a bus that took forever. Is that bad luck, or exactly what a random process predicts? Count, time and test.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumStatistics and probability
AA 4.8, 4.14 HL, 5.11AI 4.8, 4.11, 4.17 HL, 4.18 HL
Angles, trajectories, scoring systems, ratings and strategy in the sports you play or watch.
An archer aims at the inner gold ring, 12 centimetres across, 70 metres away, and the arrow leaves the bow pointing only a few degrees above the horizontal. How much can that angle wobble before the arrow misses?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumCalculus
AA 2.7, 3.5, 3.6, 5.6, 5.9, 5.14 HL, 5.18 HLAI 2.10 HL, 5.6, 5.9 HL, 5.13 HL, 5.16 HL
Tried by our studentsFull guide
A rating is a number that says how likely you are to win. Every game nudges it up or down. How does one formula keep track of a whole league's strength?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AA 1.5, 1.7, 2.2, 2.5, 2.9, 4.9, 5.6AI 1.5, 2.2, 2.5, 2.9 HL, 4.9
Every coach says shoot with more arc. A steeper shot has a bigger hole to aim at, but it needs more speed. So where does the best angle really sit?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingCalculus
AA 3.5, 3.6, 3.10 HL, 4.9, 5.6, 5.8, 5.14 HLAI 5.6, 5.9 HL, 4.9
Tried by our studentsFull guide
A penalty is over in less than a second, so the keeper has to move before seeing the ball. If the kicker always shoots to one side, the keeper learns it. So how often should each of them go left?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumStatistics and probability
AA 2.1, 4.5, 4.7, 4.11AI 4.5, 4.7, 4.11, 4.6
Win 55% of the rallies and you would expect to win about 55% of the games. You will not. You are going to find out how far off that guess is.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumStatistics and probability
AA 1.3, 1.8, 1.9, 4.5, 4.8, 5.6AI 1.3, 4.5, 4.8, 1.11 HL, 5.9 HL
Tried by our studentsFull guide
Commentators say a player is "on fire". You're going to ask whether that is real or just what randomness looks like, and let a p-value settle the argument.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumStatistics and probability
AA 4.5, 4.6, 4.8, 4.11AI 4.5, 4.8, 4.11, 4.18 HL, 4.19 HL
Everyone knows the best angle for a long throw. But the pitch is rarely flat: shot put, free throw, throwing off a wall. So does the rule survive?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumCalculus
AA 2.7, 3.5, 3.6, 5.6, 5.8, 5.9, 5.14 HL, 5.18 HLAI 5.6, 5.7, 5.13 HL, 5.16 HL
Two teams finish on the same points. Who lifts the trophy? You're going to find out whether the tie-break rule quietly decides the league.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumStatistics and probability
AA 4.5, 4.7, 4.8AI 4.5, 4.7, 4.8, 4.17 HL
Trigonometry, vectors and geometry on the Earth and beyond: navigation, maps, orbits and shapes.
Anyone can write H = d tan θ. The real question is how far you can trust the answer, and which measurement lets you down first.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleGeometry and trigonometry
AA 3.2, 3.3, 5.1, 5.6AI 1.6, 3.2, 3.3, 4.3
Near the equator, day length barely changes all year. Near the Arctic Circle, it swings from almost nothing to almost everything. One equation explains both, and the tilt of the Earth is in it.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AA 2.10, 3.4, 3.5, 3.7, 3.8, 3.9 HL, 5.6AI 2.5, 2.6, 3.8 HL, 4.4, 4.13 HL
Your phone doesn't know where it is. It only knows how long signals took to arrive from points it can't see. From those times it finds a spot on the Earth, and the geometry of the beacons decides whether that spot is good.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingGeometry and trigonometry
AA 1.16 HL, 2.10, 3.1, 3.12 HL, 3.13 HLAI 1.8, 1.14 HL, 3.1, 3.5, 3.10 HL, 4.3
On a flat map, the shortest way from Asia to Europe looks like a straight line. On the globe it isn't, and the flight that looks curved is the shorter one. How much shorter, and where does it matter most?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumGeometry and trigonometry
AA 3.1, 3.2, 3.4, 3.5, 3.12 HL, 3.13 HLAI 3.1, 3.2, 3.4, 3.7 HL, 3.10 HL, 3.13 HL
Everyone says bees are brilliant mathematicians. You're going to test the claim and find out what is actually true.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleGeometry and trigonometry
AA 3.1, 3.2, 3.4, 5.1, 5.8AI 3.2, 3.4, 5.1, 5.7
Greenland looks as big as Africa on many maps. You're going to put numbers on exactly how wrong that is, and why.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingGeometry and trigonometry
AA 3.4, 3.5, 3.9 HL, 5.6, 5.10, 5.16 HLAI 3.4, 3.7 HL
Kepler spent years on Mars's orbit. You're going to fit an ellipse to real positions and find out how good the fit really is.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingGeometry and trigonometry
AA 2.10, 3.2, 3.4, 3.5, 3.7, 3.8, 5.8AI 1.6, 4.13 HL
A regular pentagon can be drawn with only a ruler and a compass. A regular heptagon cannot. The reason sits inside the complex numbers, and you're going to find it.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingAlgebra and functions
AA 1.12 HL, 1.13 HL, 1.14 HL, 1.15 HL, 2.12 HL, 3.5, 3.10 HLAI 1.13 HL
GPS satellites carry clocks that gain about 38 microseconds a day compared with clocks on the ground. That tiny gap comes from a square root, and the square root is almost always replaced by a short series. How short can it be?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingCalculus
AA 1.9, 1.10 HL, 1.8, 5.1, 5.6, 5.13 HL, 5.19 HLAI 1.1, 1.6
Tried by our studentsFull guide
Fitting models to real data from science and society: growth, climate, technology and populations.
A ball bounces for ever in the maths and stops in your hands. You're going to measure the gap between those two stories.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleModelling
AA 1.3, 1.5, 1.8, 2.9, 1.15 HLAI 1.3, 1.5, 2.5, 1.11 HL, 2.10 HL
Since 1958 one mountain in Hawaii has recorded the carbon dioxide in the air, and the graph has a zigzag on top of a climb. Which function describes the climb, and what year does each model say we hit 450 ppm?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AA 2.6, 2.9, 3.7, 4.4AI 2.5, 2.6, 4.4, 4.13 HL, 5.9 HL
Some countries have fewer than one birth per woman. Nothing dramatic happens in year one, but a matrix can show you what the age structure looks like in fifty years, and why it cannot be fixed quickly.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingModelling
AI 1.14 HL, 1.15 HL, 3.15 HL, 4.19 HL, 2.6, 1.3, 4.12 HLAA 1.16 HL, 1.3
Every two years or so, the number of transistors on a chip is said to double. That is a claim you can test with a ruler and a log scale, and you can look for the point where it stops being true.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AA 1.5, 1.7, 2.9, 4.4AI 1.5, 2.5, 2.6, 2.10 HL, 4.4, 4.13 HL
Any spreadsheet will draw a curve through population data. The real question is which model deserves to, and what happens when you use it to predict.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AA 1.5, 2.9, 4.4, 5.18 HLAI 2.5, 2.9 HL, 4.4, 4.13 HL, 5.14 HL
Curves, fractals, musical pitch and the shapes of buildings and fonts.
Software will fit a cubic, a quartic, a degree-9 monster to your arch and give you a lovely R². The real question is which one deserves belief.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleAlgebra and functions
AA 2.6, 2.12 HL, 4.4, 5.11AI 2.5, 2.6, 4.4, 4.13 HL, 5.5
The letters on this page are made of curves, and each curve is stored as just three or four points. You're going to find out how four points make a smooth curve.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumGeometry and trigonometry
AA 1.9, 2.6, 3.4, 5.3, 5.5, 3.12 HL, 3.13 HLAI 2.5, 3.10 HL, 5.3, 5.8
A picture is just a set of curves, if you are patient enough to write them down. You're going to rebuild your own drawing from functions, and then find out how close you really got.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleAlgebra and functions
AA 2.1, 2.6, 2.11, 3.7, 5.11, 1.16 HLAI 2.1, 2.5, 2.8 HL, 2.9 HL, 1.8, 5.8
Tried by our studentsFull guide
A flute and a violin can play the same note and sound completely different. You're going to find the maths of the difference.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingGeometry and trigonometry
AA 3.6, 3.7, 3.10 HL, 5.11, 5.16 HL, 5.19 HLAI 3.8 HL, 5.11 HL
Some people see the golden ratio in everything. You're going to measure, and find out whether it is really there.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleGeometry and trigonometry
AA 2.7, 1.8, 4.3, 1.15 HLAI 1.6, 4.3, 4.11, 4.16 HL
Square a number, add a constant, repeat. For some constants the numbers blow up, for others they stay put, and the border between them is infinitely detailed. You're going to find part of that border exactly.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingAlgebra and functions
AA 1.12 HL, 1.13 HL, 1.14 HL, 1.15 HL, 2.7, 4.5AI 1.12 HL, 1.13 HL, 4.5
Tighten a guitar string and the note rises. But by how much? You're going to turn a tuner app into a data logger, and let the data pick the function.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleAlgebra and functions
AA 1.7, 2.9, 2.10, 4.4, 5.3AI 1.6, 2.5, 2.10 HL, 4.4, 4.13 HL
Strike a drum and you hear one note, but the head is vibrating in many ways at once. You're going to test what sets the lowest note and why a small weight at the centre changes it.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumModelling
AA 1.7, 2.8, 2.9, 5.3, 5.7, 5.11AI 2.5, 2.10 HL, 4.4, 4.13 HL, 5.8
Tried by our studentsFull guide
Approximation, numerical methods, number theory and the maths behind codes and computers.
Computers use base 2. We use base 10. Babylonians used base 60. Which one is best? You're going to find that the answer depends on what you count as cost.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
GentleAlgebra and functions
AA 1.5, 1.7, 2.2, 5.3, 5.6, 5.8AI 1.5, 5.6, 5.7
Every barcode ends in a digit that checks the others. Swap two digits and the scanner usually shouts, but sometimes it stays quiet. You're going to prove exactly when it stays quiet.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumAlgebra and functions
AA 1.6, 1.15 HL, 4.5, 4.6AI 1.6, 4.5, 4.6
Join points on a circle and count the regions. 1, 2, 4, 8, 16... you think you know what comes next. Draw one more point and check.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumAlgebra and functions
AA 1.2, 1.6, 1.9, 1.10 HL, 1.15 HL, 2.12 HL
How many ways can you pay 50 cents? By hand it is easy, then hopeless. You're going to build a machine that counts for you, then ask how fast the count explodes.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumAlgebra and functions
AA 1.3, 1.8, 1.10 HL, 1.11 HL, 1.15 HL, 5.19 HLAI 1.3, 1.11 HL
Your calculator never 'knows' the value of sin 0.3. It builds it from a handful of terms. How many does it need, and how do you know?
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumCalculus
AA 5.19 HL, 5.13 HL, 5.6, 5.7, 1.8, 5.12 HL
Your calculator finds roots in a blink. Newton's method is the idea inside. You're going to test how fast it works, and then go looking for the starting point where it breaks.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
MediumAlgebra and functions
AA 5.4, 5.6, 2.10, 5.12 HL, 5.13 HL, 1.12 HLAI 5.4, 5.9 HL
1/7 gives 0.142857142857... and then it repeats, forever. Why that block, why 6 digits, and what would 1/13 do? You're going to find out why, not only what.
AA SL ●●● HL ●●●AI SL ●●● HL ●●●
DemandingAlgebra and functions
AA 1.3, 1.6, 1.8, 1.15 HLAI 1.3, 1.11 HL
Help with your Maths IA
Our Maths students get IA support as part of tuition: help to choose an aim, find a twist, plan the exploration and improve the draft. We give feedback, questions and teaching. We never write or rewrite your IA.
Free trial lesson · small groups · 5 Tank Road or online
The Maths IA is an individual written exploration, marked out of 20 and worth 20% of your final grade. It is the same task at SL and HL, in AA and AI. The report is about 12 to 20 pages, and the IB suggests about 10 to 15 hours of teaching time.
A good topic has a clear, narrow aim, maths at the level of your course, and a problem that is yours rather than a textbook exercise. It should leave room to reflect as you go. A few things done properly beat many things done quickly.
Often yes, but the fit differs. Each topic shows its fit for AA SL, AA HL, AI SL and AI HL. The mathematics is judged against your own course, so the same exploration can be strong evidence in one course and routine in another.
Treat every topic as a starting point. Marks go to your own aim, your own choices and your own thinking, so a template topic rarely scores well. Use the free guide to see the set-up, then change the question, the data or the model so it becomes yours.
Gentle, medium and demanding show how much mathematics a topic asks for and how easy it is to finish well. A gentle topic done carefully can score better than a demanding one left half done, so choose a level you can complete with care.
No. We give feedback, questions and teaching. Our Maths students get help to choose an aim, find a twist and improve their draft. Anyone can also get a research question checked free with the IA and EE Coach, which says what to fix and never rewrites the question.
IB and International Baccalaureate are trademarks of the International Baccalaureate Organization. Photon Academy is not affiliated with or endorsed by the IB. The topics on this page are our own and are not taken from IB materials.
Mr Ejaz Ahmad and his team
5.0
Taught by IBO-certified tutors & practising IB examiners.
