Interactive Math & Science Lessons
Not just answers — real insight across math, science, and curiosities.
307 interactive lessons · 33 chapters that run in order · 48 articles
Short sequences rather than single tools. Each step hands the next one something specific, so they are worth doing in order.
L1: foundations
MATHFour tools that build one lesson: a summary number is a lossy compression, and how you compress determines what the summary is honest about. Each step hands the next one a specific limitation of single-number summaries.
L1: foundations
MATHSix tools that build the standard triangle and circle results in the order they actually depend on each other. Nothing here is asserted: every rule arrives as a consequence of the one before it, and the route ends by putting the first one back on a grid, where it turns into the distance formula.
L1: foundations
MATHThree tools and one argument: some numbers cannot be written out in full, the best short version of one can be searched for rather than guessed, and cutting a number short has a cost you can measure.
L1: foundations
MATHSix tools that build equation-solving in the order the ideas actually depend on each other, starting from what a minus sign does and ending with a machine method for three unknowns at once. Every step hands the next one a specific thing: a move, a restriction on a move, or a test for whether a move was legal.
L1: foundations
PHYThe ratio of voltage to current is the whole of the first tool. The second adds a component that has no resistance and limits the current anyway. The third asks what that costs at each frequency.
L2: school
MATHFour tools that build one idea: money is worth different amounts at different times, and a repeated payment can be collapsed into a single formula. Each step here hands the next one something specific.
L2: school
MATHThree tools, one habit: when quantities differ by factors rather than amounts, stop counting and start counting exponents. The first builds the ruler, the second walks it, and the third puts a number on it with no data at all.
L2: school
MATHThe birthday problem looks like a party trick and is actually the standard tool for reasoning about collisions. The second step is where computer scientists meet it again, under a different name.
L2: school
PHYTwo tools and one shortcut: when all you want is speeds and heights, you can skip the forces entirely and let energy keep the books.
L2: school
MATHFour tools, and each one exists because the tool before it runs out. Factorising works when the roots are tidy, the formula works when they are not, division works past degree two, and a rational function is what two polynomials do when one sits over the other.
L2: school
BIOOne exponential, read three ways. Forwards it tells you how much is left; backwards it tells you how old something is; and applied to DNA rather than to nuclei it needs a rock to set it by.
L2: school
PHYA lens is a boundary that somebody curved, and no page on this path introduces physics the first one does not already contain. What changes is how much is asked of it: one surface, then an ideal lens, then a piece of glass.
L2: school
PHYFive machines and one bargain. None of them creates energy, and every one of them lets you choose how to spend it: a smaller force over a longer path, or the reverse. Each step hands the next a specific piece of that bargain, and by the end you can read any machine by asking two questions about it.
L2: school
MATHThree tools, and the point of the sequence is that they are not three subjects. A sine wave is what uniform circular motion looks like from the side, and a swinging pendulum is what it looks like when gravity supplies the turning.
L2: school
ASTRONothing about a star can be worked out from how bright it looks until you know how far away it is. That is the order these four go in, and it is the order astronomy went in — each rung measured against the one below it, all the way from a surveyor’s triangle to the edge of the observable universe.
L2: school
STATSix tools and one distribution. Everything here is the binomial in a different limit, or the binomial read backwards: count the successes, let the count get large, and watch a bell curve arrive that nobody put there.
L2: school
PHYThree tools, and the third one breaks the first two. Constant acceleration has a closed-form answer, so the first two steps are algebra. Air resistance is not constant, and from there the only honest method is to integrate it step by step.
L2: school
MATHFive tools on one idea: a smooth curve, magnified far enough, is indistinguishable from a straight line. That single fact is what a derivative measures, what a Taylor series extends, and what a gradient does in more than one direction. The path starts with the angle measure that makes the arithmetic come out clean and ends on a hillside where the steepest slope beats every direction you actually measured.
L2: school
PHYTwo tools built on a single result. Water leaves a hole at exactly the speed a stone reaches falling the same depth, and everything else follows from where that speed is spent: sideways it is a projectile, downwards it is the tank emptying itself.
L2: school
STATFour tools asking one question with progressively less to work with: two groups, then any number of them, then counts with no average in them at all. The last tool asks it before the data exist.
L2: school
PHYOne restoring force, followed as far as it will go. A mass on a spring oscillates, a row of them passes the oscillation along, two of those meeting decide whether to add or cancel, and a boundary at each end throws away every frequency that does not fit.
L2: school
STATNobody objects to the arithmetic in Bayes' theorem. They object to the prior — where the first number came from, and who chose it. These three tools take that question in the order it gets harder: a prior you can look up, a prior the apparatus hands you, and a prior nobody has.
L2: school
COMPA video encoder is not one algorithm but eight stages in a fixed order, and the order is not arbitrary: each stage exists because the one before it made its job possible. This tool models one of them. The chain below links to the other seven.
L2: school
BIOEvery population here is held back by something, and the something gets less obliging at each step: first the crowding it causes itself, then a fixed catch taken off the top, then a predator that is a population in its own right, and finally running out of people left to infect.
L3: undergraduate
MATHThree tools that turn a picture of scattered points into a number you can defend. The route runs through geometry rather than statistics, because the geometry is what makes the statistics honest.
L3: undergraduate
MATHOne question, asked four times with less and less to go on: what does this matrix actually do to space? The columns answer it outright. The eigenvectors answer it for the directions that survive. The SVD answers it when there are none, and PCA points the same machinery at a cloud of measurements.
L3: undergraduate
ASTROAn orbit is fixed by its size and its shape, and almost everything else follows. These five tools go from the ellipse to the fuel: where a spacecraft is, how fast it is going there, how long the trip takes, which trip is cheapest, and what that costs in propellant.
L3: undergraduate
COMPTwo tools that open up a large language model at the moment it chooses a word. The second literally runs on the first: the candidates it ranks are entries from the vocabulary the first one builds.
L3: undergraduate
STATSix tools on the same 98 years of returns, each handing the next a specific quantity. The sequence answers one question in six steps: what is actually risky about an investment, and how much of what you are told about it can be checked?
L3: undergraduate
COMPA stopwatch measures your laptop. Counting operations measures the algorithm. Two tools built on that one habit.
L3: undergraduate
MATHNewton's method, Simpson's rule and Euler's method all run on the same engine: inside a small enough interval, treat the curve as straight. Each one therefore has a failure that is not a bug, and the useful knowledge is which failure belongs to which method. Every step here hands the next a named error and the rate at which it shrinks.
L3: undergraduate
COMPFour encoders you could run on paper, ordered by what each one can see that the last one could not. The fourth compresses nothing whatsoever, and it is the reason the first one works on text.
L4: specialist
BIOFive tools that build one argument: DNA is a real storage medium and a real computer, and both run into limits you can calculate rather than guess at. Each step hands the next a specific number.
Interactive algebra, trigonometry, and calculus tools. Visualise equations, explore roots, and build intuition through sliders.
Statistics and finance tools - distributions, probability, inference, compound growth, and investment models.
Physics simulators for mechanics, circuits, waves, and oscillation. See the equations come to life.
Orbital mechanics, rocket equations, and planetary physics. Explore the solar system and beyond.
Computing tools spanning logic gates, quantum algorithms, and video encoding - from classical to quantum.
Biology, geology, and chemistry tools - life sciences, Earth processes, reactions, and molecular properties.
The Tidal Zone - fun and pseudo-scientific tools. Biorhythms, moon phases, and more. Take results with a pinch of salt.
Everyone is told that the sample variance divides by n − 1 because a degree of freedom was consumed. That names the bookkeeping. The mechanism is that the mean was fitted to the data, and a fitted mean always makes the data look tighter — by one observation’s worth, at every sample size.
05.09.2026
Self-locking and "wastes more than half of what you put in" turn out to be the same inequality written two ways. That is why a bolt needs no ratchet, and why turning one is such miserable work.
03.09.2026
There are two moves in school algebra: add something to both sides, or multiply both sides by something. Adding has never needed a warning, and every rule you had to memorise is about the other one.
30.08.2026
