Interactive figures
Four things on this site you can actually operate rather than read. Each was built for one standard and each is useful for more than one, so they are listed here rather than left buried in the topic page they happen to live on.
Series RLC phasor diagram
Four rotating phasors and the waveform they project, on one clock. Shows why V_L leads and V_C lags, where φ comes from, and what resonance looks like.
- Reactance
- Impedance
- Phase angle
- Resonance
- AS 91524 Mechanics — the same phasor idea underlies SHM: circular motion projected onto a line.
- Any AC question in an internal or a derived-grade paper, where the marks are for the phase reasoning.
Speciation in an aqueous system
Which species dominates at which pH, with the pH marker you drag. Makes pKa the crossover point rather than a number in a table.
- Weak acids
- Buffers
- pKa
- Buffers
- pH
- AS 91391 Organic chemistry — whether an amine or a carboxylic acid is charged at a given pH.
- The titration curves in the internal: the buffer region is the flat part for the same reason.
The tangent, and the derivative it traces
Drag along a curve and watch the gradient of the tangent draw the derivative. The two graphs are the same information, which is the whole idea.
- Gradient
- First principles
- Stationary points
- AS 91579 Integration — read it right to left and the area under the derivative is the change in the function.
- AS 91577 Algebra — where a graph turns, and what that means for the sign of the gradient.
Bootstrapping a difference in medians
Two samples, both medians, and the difference between them resampled until an interval appears. Zero is drawn as a hard line, because whether the interval clears it is the call.
- Bootstrap
- Confidence interval
- Medians
- Making the call
- AS 91584 Time series and AS 91581 Bivariate — the same "how much would this move if I sampled again" question.
- AS 91586 Probability distributions — what sampling variability is, before you model it.
SHM as circular motion, seen edge-on
A point going round a circle at constant ω, and the shadow it casts on the vertical diameter. The shadow IS the oscillating mass, so v is largest in the middle and zero at the ends without any of it having to be asserted.
- SHM
- Amplitude
- Angular frequency
- Phase
- AS 91526 Electrical systems — the same projection is what a phasor diagram does, so this and the phasor figure are one idea twice.
- AS 91578 Calculus — a = −ω²x is the differential equation d²x/dt² = −ω²x, and the three stacked traces are a function and its first two derivatives.
Two-source interference and path difference
Two coherent sources and a point you drag, with both distances drawn and the extra bit of the longer path marked off to scale. Δ ÷ λ is a live number, so n stops being a label and becomes what it is: how many whole wavelengths of extra travel.
- Superposition
- Path difference
- Coherence
- Fringes
- AS 91523 diffraction gratings — the same d sinθ = nλ, with the path difference between adjacent slits.
- Any question asking WHY a fringe is dark: the answer is one sentence about path difference, and this is that sentence drawn.
Spontaneity: ΔG against temperature
ΔG = ΔH − TΔS drawn as a straight line, with ΔH the intercept and −ΔS the gradient. The four rows of the spontaneity table become one question: which way does the line lean, and where does it cross zero?
- Gibbs free energy
- Entropy
- Enthalpy
- Spontaneity
- AS 91392 Aqueous equilibria — why a process can be favoured at one temperature and not another, which is the same argument behind Kw changing with temperature.
- The calorimetry internal: ΔH measured there is the intercept of this line.
A titration curve, read properly
pH against volume added, with equivalence, half-equivalence and the indicator range marked ON the curve. Switching from a strong to a weak acid moves the equivalence point off pH 7, which is the assumption that costs the most marks on this standard.
- Titration
- Equivalence point
- pKa
- Buffers
- Indicators
- AS 91392 buffers — the flat stretch is the buffer region, and it is flat for the reason the speciation figure shows from the other side.
- The titration internal: choosing an indicator is a question this curve answers by looking.
The Argand plane and De Moivre
Drag z and watch its powers spiral: each one turned by arg z and stretched by |z|. Switch to roots and they space themselves evenly round a circle, because they have to.
- Complex numbers
- Modulus
- Argument
- De Moivre
- Roots
- AS 91577 loci — |z − a| = r is a circle for exactly the reason the modulus is a distance here.
- AS 91526 Physics — a phasor is a complex number of constant modulus and rotating argument, which is the |z| = 1 case.
Riemann sums becoming the integral
Rectangles under a curve, and the exact area from the antiderivative. Left and right sums err in opposite directions, so the answer is trapped between them — and the trap tightens every time n doubles. That is what a limit is.
- Definite integral
- Area under a curve
- Limits
- Approximation
- AS 91578 Differentiation — read the tangent figure and this one together and the Fundamental Theorem is the sentence joining them.
- AS 91584 Statistics — the area under a density curve is this same sum, which is where a probability comes from.
Standing waves: string, open pipe, closed pipe
All three cases at the same length and the same harmonic, oscillating together. The closed pipe visibly fits a quarter wavelength where the others fit a half — and skips every even harmonic as you step n.
- Standing waves
- Harmonics
- Nodes
- Resonance
- AS 91524 Mechanics — a standing wave is SHM at every point, with the amplitude varying along the string.
- The practical investigation: resonance in a tube is the closed-pipe case, and this is what you are measuring.
The photoelectric effect
A metal, a beam whose colour tracks frequency, and the Eₖ(max) against f graph. The intensity slider is there to do NOTHING to the energy — a control that visibly refuses to help is a better argument than a sentence saying it will not.
- Photons
- Work function
- Threshold frequency
- Planck's constant
- AS 91523 Waves — the case where the wave model runs out, and why light needed both descriptions.
- AS 91526 Electrical systems — the stopping voltage is a p.d. doing work on a charge, eVs = Eₖ.
Margin of error, and what a poll can claim
Two intervals on one percentage axis, with the overlap shaded. Whether you may claim a difference is a geometric question — does this bar cross that line, do these two share ground — and the verdict is given in the wording an answer needs.
- Margin of error
- Confidence interval
- Sample size
- Making the call
- AS 91582 Formal inference — the bootstrap figure builds the interval; this one is about what you may then say.
- AS 91581 Bivariate — the same "could this be sampling variation" question about a relationship.
The normal distribution, as an area
A bell curve with the region you are asking about shaded, and the same answer shown as an area, a z-score and the calculator call. Includes the inverse question, which is a different button and a different sentence.
- Normal distribution
- z-score
- Inverse normal
- 68-95-99.7
- AS 91585 Probability — an area under a density curve is where a continuous probability comes from at all.
- AS 91579 Calculus — that area is a definite integral, which is what the Riemann figure is about.
Punnett squares: monohybrid, sex-linked, dihybrid
A real grid, not a canvas drawing of one — pick complete, incomplete or co-dominance on the same cross and watch the SAME 1:2:1 genotype ratio give a different phenotype ratio each time. The sex-linked tab makes it visible that Y carries no colour allele at all, which is the whole reason a tortoiseshell cat cannot be male. The dihybrid tab reveals its 16 cells on request, so 9:3:3:1 is counted rather than quoted.
- Genotype ratio
- Phenotype ratio
- Co-dominance
- Sex linkage
- AS 91156 Life processes — meiosis is where independent assortment happens; this figure is what it produces.
- AS 91159 Gene expression — the same dominant/recessive vocabulary, one level down at the allele-to-protein step.
Consumer surplus, producer surplus, and deadweight loss
One demand-and-supply graph, five switchable interventions (none, a tax, a subsidy, a price ceiling, a price floor). Consumer surplus, producer surplus, the government revenue-or-cost rectangle and the deadweight-loss triangle are all shaded from the actual polygon geometry, worked live from the slider — never typed in — and a dashed line marks the shortage or surplus a price control creates.
- Consumer surplus
- Producer surplus
- Deadweight loss
- Price controls
- AS 91400 Market structures — the SAME deadweight-loss idea reappears where a monopoly sets P above MC instead of a control setting price directly.
- AS 91403 Macro-economics — a tariff or quota question is this exact graph with the wedge relabelled.
MR = MC: perfect competition vs monopoly
One firm’s MC and AC curves, with a segmented control switching its demand between a flat price-taker line and a downward-sloping monopoly AR. The SAME MC = MR rule sets quantity either way — the figure never special-cases the mode — and price is always read off AR, the exact step marginal-analysis warns is skipped. Shades the profit-or-loss rectangle and, only when AR is sloped, the deadweight-loss triangle between the monopoly outcome and where a price taker facing the same demand curve would have landed.
- Marginal revenue
- Marginal cost
- Allocative efficiency
- Deadweight loss
- AS 91399 Market equilibrium — the same deadweight-loss idea from the other direction: a control moving price directly instead of a firm choosing where P > MC.
- AS 91403 Macro-economics — the marginal-analysis reading-price-off-demand trap recurs in any profit-maximisation question.
Ray diagrams for lenses and mirrors, and Snell’s law
Two candidates merged into one figure rather than shipped separately, since they are two sections of the same topic. The lens/mirror tab draws the SAME two construction rays a student draws by hand for all four elements (converging/diverging lens, concave/convex mirror) and finds the image wherever they actually cross — real if that crossing is on the physical side, virtual (dashed backward extension) if not. The refraction tab computes θ₂ directly from Snell’s law and simply stops drawing a refracted ray once sinθ₂ would exceed 1, which IS total internal reflection rather than a separate rule bolted on.
- Ray diagrams
- Real vs virtual
- Snell's law
- Total internal reflection
- AS 91173 Electricity — the same "trace the actual construction rather than recall a diagram" habit applies to circuit diagrams.
- AS 91523 Wave systems (L3) — refractive index and wave speed reappear wherever a wave crosses a boundary.
Series, parallel, and a fault mid-circuit
The same three resistors and battery, switchable between all-series, all-parallel, and the realistic mixed circuit (R1 in series with R2 ∥ R3) every paper actually draws — with a real fault switch on R2 and R3, not a slider quietly sliding to zero, so opening one genuinely removes it rather than short-circuiting it. Makes circuit-analysis-every-paper’s own mistake callout checkable rather than assertable: set internal resistance r to 0 and a parallel branch failing moves the OTHER branches’ readouts by exactly nothing; raise r and it moves them a little.
- Series circuits
- Parallel circuits
- Circuit faults
- Internal resistance
- AS 91170 Waves — the same "trace the actual construction" discipline as the ray-diagram figure, applied to current paths instead of light paths.
- AS 91526 Electrical systems (L3) — the DC series/parallel rules this figure drills are the DC special case of the AC phasor figure’s impedance rules.
A launch, split into its two components
One trajectory and two stacked strip-charts beneath it — vₓ(t), a flat line nothing after launch ever changes, and vᵧ(t), the one straight line ticking through zero at the apex. A time slider (or a Fly button) parks the flight at any instant and shows both velocity components as arrows at once, which is what a "without calculating, compare" question actually needs — not a moving picture, a frozen one you can put exactly where the question is asking about.
- Projectile motion
- Independent components
- Time of flight
- Range
- AS 91524 Mechanical systems (L3) — the same horizontal/vertical decomposition, with the vertical component becoming a full SUVAT problem rather than a level launch.
Substitution, insertion, deletion — on a real sequence
A real DNA template strand (the site’s own worked example, extended to five codons), with the full 64-codon genetic code behind it. Substitute a base and only its own codon’s amino acid can change — classified as silent, missense or nonsense from the real table, not asserted. Insert or delete one instead and every codon after that point re-derives from scratch, since mRNA and the amino-acid chain are ENTIRELY recomputed from the current sequence on every edit — a frameshift is simply what that recomputation produces once the length has changed, not a separate rule.
- Substitution
- Frameshift
- Genetic code
- Silent vs missense vs nonsense
- AS 91156 Life processes — the same DNA→RNA→protein pipeline, one level up from where a mutation happens.
- AS 91157 Genetic variation — mutation is the source of the NEW alleles a Punnett cross then inherits.
Osmosis: the same solution, two different cells
One slider — the solute concentration outside the cells, against a fixed 30% inside — drives an animal cell and a plant cell side by side. Water direction is identical for both; the wall is what differs, so the animal cell swells, bursts or crenates while the plant cell turns turgid, flaccid or plasmolysed. Solutes never cross, water drifts across at a rate set by the difference and stops at zero, and the readout uses the named states a mark scheme asks for rather than a number.
- Osmosis
- Tonicity
- Turgor
- Plasmolysis
- AS 91155 Adaptation — root hairs and guard cells work because of exactly this water-potential logic.
- AS 91160 Microscopic investigation — the plasmolysed onion cell is the standard slide for this idea.
Shape and polarity from electron regions
One rotatable molecule with segmented controls for electron regions (2–4) and lone pairs. Electron-region geometry and molecular shape are separate readouts on purpose — add lone pairs to a fixed 4 regions and the first never moves while the second does, which is the parent-versus-final distinction the section warns about. Each bond is a real dipole vector and the net dipole is their 3D sum, so CO₂, BF₃ and CH₄ cancel because the vectors do, NH₃ and H₂O do not, and swapping one outer atom is CCl₄ versus CH₃Cl. Directions are drawn at the ideal geometry; the true squeezed angle is printed.
- VSEPR
- Molecular shape
- Lone pairs
- Polarity
- AS 91165 Organic — the shape of every carbon centre (tetrahedral, trigonal planar, linear) is this table.
- AS 91391 Organic compounds (L3) — shape and polarity decide the intermolecular forces the property arguments rest on.
Energy profile and particle energies, together
A reaction profile above a distribution of particle energies, sharing one Eₐ. Add a catalyst and the peak drops while reactants, products and ΔH stay exactly where they were — the commonest profile mark lost is moving the products — and in the lower panel the Eₐ marker moves left so more of the distribution lies beyond it. Raise the temperature instead and the marker stays put while the curve flattens: the same extra area, from the other direction. The lower panel is labelled schematic, since at real activation energies the fraction beyond Eₐ is far too small to draw to scale.
- Energy profile
- Activation energy
- Catalyst
- Temperature and rate
- AS 91164 Bonding and energy changes — the same labelled profile, where the marks are for ΔH and Eₐ as separate arrows.
- AS 91390 Thermochemical principles (L3) — ΔH and the size of Eₐ are still independent quantities there.
Break-even, margin of safety and target profit on one graph
Revenue and total cost as two lines on one set of axes, so the three numbers every paper asks for become three positions on one picture: where the lines cross (break-even), how far the expected-sales marker sits to the right of that (margin of safety), and how far right you must go to clear a profit target. Round-up is shown when it happens, a capacity line marks the edge of the relevant range and a target beyond it is reported as NOT achievable, and a non-positive contribution margin says there is no break-even instead of dividing by it.
- Contribution margin
- Break-even
- Margin of safety
- Target profit
- AS 91404 Accounting concepts — the same fixed-versus-variable distinction, one level up from the decision it feeds.
- AS 91400 Market structures (Economics) — fixed and variable cost shape the cost curves there for exactly this reason.
The Doppler effect: a moving source and a moving observer
Wavefronts drawn from a moving source, and separately an observer running through the wavefronts of a still one. With a moving source each ring is centred where the source was when it made it, so rings bunch up ahead and spread out behind — the wavelength really changes. With a moving observer the rings stay concentric and the spacing does not change; the observer just meets them faster. That is why the source term sits on the bottom of the formula and the observer term on top. The speed of sound is printed as unchanged throughout, since the schedules award a mark for saying so every year. Rings are emitted slowly enough to see but every ratio drawn is the real one.
- Doppler effect
- Wavefronts
- Observed frequency
- Speed of sound
- AS 91170 Waves (L2) — the same wave-speed relation v = fλ, with the Doppler shift as what happens when the source moves.
- AS 91523 beats and interference — a frequency shift is what turns two close frequencies into an audible beat.
How the work grows — and what it costs in time
Six growth rates (log n, n, n log n, n², 2ⁿ, n!) on one chart, with every step count converted to the time it takes at a billion steps a second and stated as a person would say it — microseconds, minutes, years, or "times the age of the universe". A step count alone cannot make an exponential feel different from a polynomial; the time column does. A linear-scale view shows the exponential and factorial curves running off the chart, which is what intractable looks like. Steps are computed exactly (factorials by summing logarithms); the only approximation is how very large numbers are displayed.
- Big-O
- Time complexity
- Tractability
- Exponential growth
- AS 91898 Computer science concept (L2) — the same complexity and tractability ideas at the level the Level 2 paper asks for them.
- AS 91906 Complex programming techniques — choosing a data structure or algorithm is choosing a row of this table.
A race down a ramp, and where the energy goes
A sliding block, a solid sphere, a solid disc and a hoop released together. All four lose the same gravitational PE, so every energy bar runs to the same total — but the bar is split between translational ½mv² and rotational ½Iω² in the proportions 1 : k, so the more of it that goes into spinning the slower the body reaches the bottom. A mass slider changes every energy and no time, which is the claim that m and r cancel made visible rather than asserted. This is the reasoning the real 2023 (rolling barrel) and 2024 (flywheel driven by a falling ball) questions turn on.
- Rotational inertia
- Rolling without slipping
- Energy conservation
- Kinetic energy split
- AS 91171 Mechanics (L2) — the sliding block is the Level 2 ramp problem (mgh = ½mv²); this figure shows what changes when the body is allowed to spin.
Can a molecule be turned to fit its mirror image?
A tetrahedral carbon beside its mirror image, with the left-hand molecule turned through each of the twelve orientations a rigid tetrahedron can take (turning allowed, reflecting not) and every position compared with the mirror image — green ring where the same group sits in the same place, orange where it does not. For four different groups (lactic acid, cysteine, butan-2-ol) no orientation matches more than two, so the pair are non-superimposable: enantiomers. For propan-2-ol, with two identical groups, one orientation matches all four and the figure says the two are the same molecule. The word “non-superimposable” is the outcome of a search over every possibility rather than a claim.
- Chirality
- Enantiomers
- Optical isomers
- Non-superimposable
- AS 91165 Organic (L2) — the same tetrahedral carbon and the same "four different groups" test, before mirror images are named.
- AS 91388 Spectroscopic data — enantiomers give identical spectra, which is why optical rotation is needed to tell them apart.
How to get anything out of them
Move one control and predict what will happen before you look. A figure you watch is a video; a figure you predict against is retrieval practice, which is the only kind of studying that reliably works. Each one prints its numbers, so you can check the prediction.