The Organic Test Bench

 An interactive organic chemistry bench for NCEA Level 2 and Level 3. No quizzes, no scores, no login — just the reactions, running.

Every year I watch students learn that bromine water goes colourless with an alkene, and every year a good number of them write exactly that in an exam and get nothing for it. Not because the chemistry is wrong, but because "it went colourless" is half an observation. What colour was it before? How fast? Did it need light? Those are the marks.

The trouble is that the only way to really learn this is to see it happen, and there is a limit to how many times you can set up bromine water for a class of thirty. So I built something to fill the gap between the textbook and the fume cupboard.

The Organic Test Bench

Open the Organic Test Bench

It is a single web page with six modules. It runs in any browser, needs no login, and works on a phone as well as a projector. Nothing is marked and there is no score anywhere in it — that is deliberate, and I will come back to why.

1. The test bench

A shelf of sample bottles on one side, a rack of reagents and tests on the other. Pick one of each and the tube reacts: colours change, bubbles rise, precipitates settle, a flame burns clean or smoky, a silver mirror plates the glass.

Every result comes with three things — what you saw, what it tells you, and the equation. The observations are written the way they should appear in an answer, colour before and colour after, with the conditions that matter. At Level 2 that is eleven compounds against thirteen tests; at Level 3 it grows to sixteen against seventeen.

2. The unknown sorter

This is the module I am most pleased with. Every bottle on the shelf, unlabelled. You run a test, choose the result you got, and watch the field of candidates shrink.

The point it makes is one students routinely miss: you are not looking for a reaction, you are looking for a split. A test that every remaining candidate passes tells you nothing at all. So each test carries a bar showing how evenly it divides what is left, and tests that cannot divide anything are greyed out and marked "no split". Choosing well is the skill being assessed, not remembering that bromine water goes colourless.

There is also a "set the unknown" mode for the front of the room: you pick the bottle before the class is looking, they nominate the tests, and the app reports what that bottle would actually give. It turns into a genuinely noisy fifteen minutes.

3. The reaction map

Every conversion in the standard on one page. Tap an arrow for reagents, conditions and a balanced equation; tap a family to light up everything it does; filter by reaction type. Filtering by substitution makes the point that the haloalkane is the hub of Level 2 without anyone having to say it.

Reagents are written in the forms NZQA uses — H₂/Pt, H₂O/H⁺, aqueous KOH — with the equally valid textbook alternatives noted underneath rather than on the arrows, since answers are marked against the former.

4. The route finder

Exam questions rarely ask for one reaction. They ask you to get from an alkane to an amine and to give every reagent along the way. Pick a starting family and a target and it works out the shortest routes from the map itself, with conditions at every step.

My favourite part is what happens when there is no route. Ask it to turn an alkene into an alkyne, or to get anything at all back out of a plastic bag, and it explains why not. "Not possible with the reactions in this standard" is a real answer to a real question, and a much better one than inventing an arrow.

5. The addition bay

Markovnikov's rule, live. Choose an alkene and a reagent and you get both possible carbocations with stability bars, both named products labelled major and minor, and the symmetrical cases where the rule simply does not apply. Underneath, a panel that builds the repeating unit of an addition polymer, and at Level 3 a second one contrasting it with condensation polymerisation.

6. The oxidation ladder

Primary, secondary, tertiary. The rungs reveal in sequence with the dichromate tube changing colour alongside, and the primary track carries small sketches of distillation and reflux apparatus — because the apparatus is what decides whether you keep the aldehyde or push on to the acid, and that is exactly the bit students leave out.

Both levels, one switch

There is an L2 / L3 toggle in the header. Level 2 is AS91165; Level 3 is AS91391, and switching adds five more bottles — an aldehyde, a ketone, an ester, an amide and an acyl chloride — along with 2,4-DNPH, Tollens', Fehling's, and warming with alkali to test the gas. The reaction map roughly doubles, picking up the derivatives, hydrolysis and reduction.

I built the switch because I teach a mixed class: a Level 2 group with several students also sitting Level 3 organic. Being able to flip mid-lesson so the Level 2 group sees where their map stops, and the Level 3 group sees what grows out of it, has turned out to be more useful than I expected.

Made for a projector

There is a zoom control in the header that goes up to 260%, with keyboard shortcuts, and it remembers the setting on that machine — so the classroom computer stays big and your laptop stays normal. A "Present" button hides the introductory text and goes full screen for a bit more vertical space on a 16:9 projector.

Why there are no quizzes

This was a deliberate decision and I want to be upfront about it. There is no scoring, no streak, no "well done!". Every module is an explainer.

Partly this is because there is no shortage of drill available already. Mostly it is because the moment a thing keeps score, students start optimising for the score rather than for understanding, and they stop being willing to be wrong out loud. Almost everything worth learning in this topic comes from making a prediction, being wrong, and finding out why — and that only happens when nothing is counting.

So the app is the laboratory, and the thinking happens on paper next to it.

A note on what it is not: the reagents and conditions are written for teaching and for exam answers, not as a laboratory method. Nothing here is a substitute for actually doing the practical work where you can.

If you use it with a class I would genuinely like to know how it goes, and especially where it is wrong. It has already been through several rounds of correction — a sooty flame is not a test for unsaturation, a pH figure means nothing without a concentration, and two of the compounds originally could not be told apart by any test on the bench, which rather undermined the module built around telling them apart. All of those came from using it and noticing. There will be more.

Captain Calculus

NCEA Level 3 Calculus 2016 | AS91578 Differentiation Full Exam Walkthrough

 In this video, I work through the 2016 NCEA Level 3 Calculus differentiation exam (AS91578 – Apply differentiation methods in solving problems) step by step.

We look at how to approach the questions, how to show the working needed for NCEA, and how the more demanding Merit and Excellence problems can be tackled. Topics appearing in this exam include: basic differentiation the chain rule product and quotient rules trigonometric differentiation exponential and logarithmic functions parametric differentiation gradients and tangents stationary points second derivatives and points of inflection interpreting derivatives related rates of change optimisation maximum and minimum problems setting up equations from unfamiliar situations mathematical proof using differentiation This video is useful if you are studying NCEA Level 3 Calculus, preparing for AS91578, revising differentiation before the external exam, or working through NZQA past papers. A good way to use the video is to attempt each question yourself first, then pause and compare your approach with the worked solution. 📚 Standard: AS91578 📈 Topic: Differentiation 🎓 Level: NCEA Level 3 Mathematics / Calculus 📝 Paper: 2016 NZQA examination ⭐ Credits: 6 External Credits For more NCEA maths, statistics, physics and chemistry tutorials, exam walkthroughs and revision resources, subscribe to Captain Calculus. More NCEA resources are also available at: Captain Calculus / NCEA Help @Captain_Calc

Premier League Winter Transfer App--The window shut. The ledger didn't.

Window closed · 2 Sep 2026

The window shut. The ledger didn't.

Twenty clubs, eleven weeks, and 542 separate reasons to lose track of who plays for whom. We built a single page that keeps count so you don't have to.

£3,529mSpent on fees

£2,119mRecouped in sales

542Movements logged

£125mFernandez, the record

Somewhere around deadline day, every transfer window stops being a story and starts being a spreadsheet. Sandro Tonali goes one way, Nico Gonzalez goes the other, and by the time Chelsea have finished their 44th piece of business you've lost the thread entirely. That's the moment this dashboard was built for: one page, every confirmed deal from 15 June to the 23:00 BST cut-off on 1 September, kept straight so you don't have to hold it in your head.

It isn't a highlights reel. It's a ledger — searchable, sortable, and built to answer the question you actually have, whichever one that is.

OPEN THE LEDGER →

Who actually spent the money

Total outlay across the league passed £3.5bn this summer, but the number that matters is never the total — it's who's carrying it. Liverpool posted the heaviest net spend in the league; Chelsea, despite landing the record fee themselves, ended up the biggest net sellers once Enzo Fernandez's outgoing £125m was counted against everything they brought in.

Net spend, selected clubs

Liverpool

£217.4m out

Ipswich Town

£183.3m out

Tottenham

£166.4m out

Crystal Palace

£34.3m in

Chelsea

£48.2m in

That's one view. The dashboard has three others sitting right next to it, built from the same 542 rows.

The ledger

Every deal, filterable by club, player, or deal type — permanent, loan, free, undisclosed, swap, release, retirement. Switch between a club-by-club view and a pure cost ranking with one click.

The trading network

Deals between two Premier League clubs happen in both directions on paper. This view draws the actual traffic between them, so the Newcastle–Arsenal axis — Bruno Guimaraes one way, plenty coming back the other — is visible at a glance instead of buried in two separate club pages.

Stats lab

For the deal-obsessed: a spend-versus-recouped scatter plot with a least-squares fit, a bootstrap resampler for fee distributions, a Lorenz curve and Gini coefficient for just how unequal the spending really was, and a higher-or-lower guessing game built entirely from real fees — including, if you're brave, an undisclosed one.

What each line of the pitch cost

Every player tagged by position, so you can see in one panel whether the league spent this summer on centre-backs or wingers — toggle between broad lines (defence, midfield, attack) and the full breakdown by role.

It's one page. No login, no tracking — open it and every number is already there.

Built to be checked, not just believed

Every fee is the figure reported at the time of announcement, generally inclusive of add-ons clubs never confirm — so treat every number as a best public estimate, not gospel. Undisclosed fees, loan payments and wages sit outside every total on the page. Where a deal happened between two Premier League clubs, it appears once on each side's books, which is exactly why the trading-network view exists: to stop that double-counting from hiding the real picture.

Small things, added because they were useful rather than because a checklist demanded them: a dark and light theme that remembers your choice, a permalink button that captures your exact tab and filters in the URL so you can send someone the precise view you're looking at, and a print stylesheet for the one person who wants this on paper.

Who it's for

• Anyone who wants to know what their club actually spent, net of sales, without doing the arithmetic themselves
• Anyone settling an argument about who "won" the window
• Anyone who wants to see the whole league's business laid out at once, not doled out across forty separate live blogs

The transfer window is the most-covered, least-summarised event in football. Hundreds of individual deals, reported individually, rarely add up to anything you can actually look at. This is what it adds up to — three views on the same 542 rows, however you'd rather see them.

OPEN THE LEDGER →

Updated through deadline day, 1 September 2026

Apps for Probability Distributions NCEA 3 Statistics 91586

Here's the apps that I'm going to be using for the NCEA 3 exam 'Apply probability distributions in solving problems'. 

They cover the concepts and the five distributions: 

• Normal Distribution
• Poisson Distribution
• Binomial Distribution
• Uniform Distribution
• Triangular Distribution

The first one is a really simple, streamline online calculator: 

and the other is a self teaching app that students can use themselves.

You can download them here and here

Half Life Calculator

 Does what it says on the packet. 

Goes with this video.

Download

Nuclear Power Plant Sim for NCEA 2 and 3 Physics

 

From split atom to spinning turbine: a nuclear power station simulator for NCEA Physics

Download

A nuclear power station is, at heart, a very elaborate way of boiling water. That sounds like a joke, but it's the honest truth and it's a genuinely useful thing for students to realise: strip away the reactor and the radiation, and the back half of the plant is the same steam-and-turbine machinery as a coal station. What's remarkable isn't the electricity generation — it's the source of the heat. This simulator lets students follow the energy all the way through, from a splitting uranium nucleus to power flowing onto the grid, and then break the plant in instructive ways.

It's a single self-contained web page: open it in a browser, put it on the projector, and it runs — the plant starts up already producing power, and you drive it from a control room. Below I'll walk through the physics it shows, then flag some ways to use it in class.

Following the energy

The clearest way to understand a power station is to trace the energy through it, and the simulator is laid out as one continuous chain you can watch flowing:

Nuclear → thermal. In the reactor core, uranium-235 nuclei undergo fission, and the energy released heats water in the primary loop. This water is kept under enormous pressure — around 155 bar — so that even at over 300°C it stays liquid rather than boiling. You can watch the core glow with fission power and the hot water flow out of it, its colour tracking its temperature.

Thermal transfer. The hot primary water flows to a heat exchanger, also called a steam generator. Here it passes its heat to a separate body of water in the secondary loop, boiling it into steam. In the simulator you can see the bubbles rising as this second water boils.

Thermal → kinetic. The steam rushes to a turbine and pushes its blades, spinning them — you can see the rotors turning at a rate that tracks the actual steam flow.

Kinetic → electrical. The spinning turbine turns a generator, which produces the electricity that flows out along the transmission line to the grid.

The used steam is then cooled back into water in the condenser and pumped around again, while waste heat is carried off through the cooling tower as that characteristic plume. So the full energy story is: nuclear → thermal → kinetic → electrical, and the simulator labels each transformation as you watch it happen.

Why two separate loops?

The heat exchanger is the design feature students most often find puzzling — why not just send the reactor's water straight to the turbine? The answer is a nice piece of applied reasoning, and it's about safety, not efficiency.

Water passing through the reactor core becomes radioactive. If that water went on to the turbine hall, the entire turbine and generator building would be radioactive too. So the primary loop is kept sealed inside the containment building, and only its heat is allowed to cross into the clean secondary loop. The steam that reaches the turbine has never been near the core — the two loops never mix. It's a case where a whole piece of expensive engineering exists purely to keep radioactivity contained, and that's worth making explicit rather than letting students assume it's about performance.

The control room: three dials, three different physics

The plant is driven by three controls, and the reason it's worth having all three is that they do genuinely different things — this isn't one "power" knob wearing three hats.

Control rods. These are made of a neutron-absorbing material. Push them into the core and they soak up neutrons, so fewer are left to cause further fissions and the chain reaction slows. Pull them out and the reaction builds. They control the rate of fission, and so the reactor's power. In the simulator, one important detail is built in: the rods move at a realistic finite speed rather than snapping instantly, because a reactor that could change reactivity instantaneously would be uncontrollable.

Coolant pump. This drives the primary water around the loop. Slow it down and heat isn't carried away from the core fast enough — and something instructive happens, which I'll come to.

Grid load demand. This is how much power the grid is asking for. Turn it down and less steam is drawn off, so less energy leaves the plant, and the reactor settles to a lower power to match. It's a small illustration of how a real plant follows demand.

At normal settings the plant produces around 3,000 megawatts of thermal power and about 1,000 megawatts of electricity — the real numbers for a large pressurised water reactor.

The two lessons hiding in the failures

The controls are most valuable when you misuse them, because two of the most important ideas in reactor physics only show up when things go wrong.

Cut the coolant pump, and the reactor calms itself down. You'd expect losing cooling to send the power soaring. Instead the core temperature spikes, but the reactor power falls on its own. This is the negative temperature coefficient, and it's the single most important safety feature of a reactor design: as the core heats up, the water expands and moderates neutrons less effectively, so the reaction becomes less reactive automatically. A reactor of this kind is inherently self-stabilising — a hotter core is a less reactive core. Students can watch this happen rather than take it on trust, and it's a lovely example of negative feedback in a physical system.

Hit the emergency shutdown, and the power doesn't reach zero. The SCRAM button slams all the control rods in and fission stops within seconds. But watch the thermal power: it drops sharply, then levels off well above zero and only slowly declines. That residue is decay heat — the radioactive fission products already sitting in the fuel keep decaying and releasing energy for a long time after the chain reaction has stopped. It's a genuinely counterintuitive result and one of the most important in the topic: shutting a reactor down does not make it safe to walk away from. It still has to be cooled, for a long time. If you cut the pump and SCRAM, the simulator's warning names this as the exact failure mode behind the Fukushima accident — reactors that had shut down correctly but then lost the power needed to keep pumping coolant, and overheated on decay heat alone.

Where the energy actually goes

There's an energy bar along the bottom of the simulator that quietly makes an important point: only about a third of the nuclear energy released ends up as electricity. The rest — roughly two-thirds — leaves as waste heat through the cooling tower.

Students often assume that's just bad engineering, something a cleverer design could fix. It isn't. It's a fundamental limit set by thermodynamics. Any heat engine's maximum possible efficiency depends only on the temperatures it works between:

η_max = 1 − T_cold / T_hot (in kelvin)

With steam around 290°C and a condenser near 30°C, that ceiling is about 46%, and real friction and turbine losses bring it down to around a third. This is also why the condenser and cooling tower aren't wasteful extras — a heat engine can only do work if heat flows from somewhere hot to somewhere cold, and the condenser provides that cold end. Rejecting heat there is what allows the turbine to work at all. That's a subtle idea, and being able to point at the cooling tower while explaining it helps it stick.

A note on the physics

Underneath the diagram is a proper coupled model — reactor power, two coolant loops, and the turbine all linked together and responding to your controls, not a scripted animation. I tuned it numerically so that the steady state lands on real PWR figures (about 3,000 MW thermal to 1,000 MW electrical at roughly 33% efficiency), so that a SCRAM drops fission power about 90% within a few seconds while decay heat lingers, and so that losing coolant makes the reactor self-limit through its temperature feedback. Testing along the way caught genuine errors — an early version let a fully shut-down reactor keep generating, and another spiked past its power ceiling on startup — which is a reminder that "looks about right" and "is right" aren't the same thing.

One honest caveat: it's a lumped-parameter model, so it won't show coolant boiling away or pressure transients, and the timescales are compressed — roughly twelve times real speed — so a SCRAM or a heat-up plays out in seconds rather than minutes. The behaviour is faithful; the clock is sped up so it fits a lesson.

Using it in class

• Trace the energy chain first. Before touching a control, just follow the flow from core to grid and name each transformation: nuclear → thermal → kinetic → electrical. The plant runs on its own, so this can be pure observation.
• Ask why there are two loops. Let them puzzle over it before you explain. "Safety, not efficiency" is a satisfying answer once they've felt the question.
• Cut the pump and watch the power fall. Ask them to predict what happens to reactor power when cooling is lost. The result contradicts the obvious guess, and the negative feedback is the payoff.
• SCRAM at full power and watch for zero. It never arrives. Decay heat is the lesson, and the Fukushima connection makes it concrete rather than abstract.
• Point at the energy bar. Two-thirds of the energy is "wasted" — ask whether better engineering could recover it. The answer is no, and η = 1 − T_cold/T_hot is why.

It sits alongside the rest of the AS91172 suite, so the fission heating the water here is the same reaction that splits the nucleus in the fission simulator and runs the weapon — the same physics, harnessed instead of unleashed. Start by boiling water, and let students discover that the interesting part was never the turbine; it was where the heat came from.

Fission and Fusion Sim for NCEA 2 and 3 Physics

 

Splitting and joining: a fission and fusion simulator for NCEA Physics

Download

Here's a puzzle to open a lesson with. Splitting a heavy atom releases energy. Joining light atoms also releases energy. But splitting and joining are opposites — so how can doing a thing and undoing it both give energy out? Something has to be wrong with the question.

Nothing is. Both really do release energy, and the reason they can is one of the most elegant ideas in the whole of physics — it all comes down to a single curve. This simulator lets students run both reactions, see the energy come out of each, and then meet the curve that explains why. And on the fusion side it goes further than most classroom tools, showing not just that fusion releases energy but why it's so fiendishly hard to make happen.

It's a single self-contained web page: open it in a browser, put it on the projector, and it runs. Below I'll walk through the physics across its two reactions, then flag some ways to use it.

Fission: splitting the heavy nucleus

The first tab is fission. A slow-moving neutron drifts into a uranium-235 nucleus and is absorbed. That makes the nucleus unstable — you can watch it wobble and deform — and then it splits, here into barium-141 and krypton-92, flinging out three fast neutrons in the process.

Two things are worth drawing students' attention to. First, those three neutrons are the seed of a chain reaction: each one can go on to split another nucleus, which is the entire basis of both reactors and weapons. Second, and this is the key point, the products weigh less than what you started with. Add up the mass before — neutron plus uranium — and the mass after — barium plus krypton plus three neutrons — and there's a shortfall. That missing mass is the mass defect, and it hasn't disappeared; it's been converted into energy via E = mc².

The simulator's mass–energy ledger does this calculation live. It shows the mass before, the mass after, the difference in atomic mass units and in kilograms, and then the energy that difference corresponds to — about 173 MeV for this single fission, counting up in front of you in both joules and MeV. It's E = Δmc² turned into a running total, using the real atomic masses.

Fusion: joining the light nuclei — and why it's hard

The second tab is fusion, and it's where the tool does something classroom simulations often skip. It's easy to say "fusion joins two light nuclei and releases energy." It's much more useful to show why nature fights it so hard — because that difficulty is the reason we have working reactors splitting atoms but not yet joining them at scale.

The reaction is deuterium plus tritium — the two heavy isotopes of hydrogen — combining to make helium-4 and a neutron. But both nuclei are positively charged, and like charges repel. To get them close enough for the strong force to snap them together, they have to be hurled at each other hard enough to overcome that electrostatic repulsion — the Coulomb barrier. In the simulator you can see the repulsion pushing back as the nuclei approach, and there's a temperature slider: below about 100 million kelvin, the nuclei don't have enough energy, they're repelled, and they bounce apart without fusing. Only when you crank the temperature high enough do they finally overcome the barrier, fuse, and release their energy.

That's already a better picture than most students get. But the tool then adds the two ingredients that separate a physics demonstration from an actual reactor, through a Lawson criterion panel.

Getting one pair of nuclei to fuse isn't a power station. For a self-sustaining burn you need three things at once, and the tool gives you a dial for each:

• Temperature — enough to get individual nuclei over the Coulomb barrier.
• Density — enough nuclei packed together that collisions happen often.
• Confinement time — the plasma has to hold its heat long enough, rather than leaking it to the walls, for fusion to keep going.

Multiply those three together and you get the triple product, and there's a threshold it has to clear — the Lawson criterion. The panel computes it live, along with the probability that any given nucleus actually fuses under the conditions you've set. The lesson that emerges is genuinely hard-won: you can have a plasma hot enough to fuse, but if it's too thin or too leaky, the burn dies out anyway. Fusion is difficult not because of any one obstacle but because you have to satisfy all three conditions simultaneously.

There's a nice visible contrast with fission built into this. Fission needs no such barrier — the neutron that triggers it is uncharged, so it wanders into the nucleus with no repulsion to overcome. That's exactly why fission was harnessed decades before fusion, and why fission can sustain a chain reaction so readily. The tool lets you feel that asymmetry rather than just be told it.

The curve that explains both

Now back to the opening puzzle. How can splitting and joining both release energy? The answer is the binding energy per nucleon curve, and the simulator draws it, updating to highlight whichever reaction you're running.

The curve plots how tightly bound each nucleon is, against the size of the nucleus. It rises steeply from the light elements, peaks at iron-56, and then declines slowly through the heavy elements. Iron sits at the top — it's the most tightly bound, the most stable nucleus there is.

Everything follows from that shape. Any reaction that moves nuclei towards iron increases the binding energy per nucleon, and the difference is released as energy. There are two ways to move towards the peak:

• From the right, split a heavy nucleus like uranium into medium-sized fragments — that's fission, sliding down the right-hand slope towards iron.
• From the left, fuse light nuclei like hydrogen into heavier ones — that's fusion, climbing the steep left-hand slope towards iron.

So splitting and joining aren't really opposites in the way the opening puzzle implied. They're both journeys towards iron, from opposite ends. That's the single idea that dissolves the paradox, and once students see the curve, the whole topic clicks into a shape they can hold in their head.

The curve also explains a quantitative fact the tool highlights: fusion releases several times more energy per nucleon than fission, because the left-hand slope is so much steeper than the gentle right-hand decline. Per kilogram of fuel, fusion is the far bigger prize — which is a large part of why so much effort goes into chasing it despite the difficulty.

A note on the numbers

The energy figures come from real atomic masses: about 173 MeV per fission of uranium-235, about 17.6 MeV per deuterium–tritium fusion. The mass–energy ledger runs the actual E = Δmc² arithmetic that a student would do by hand, just instantly. On the fusion side, the temperature threshold and the Lawson triple product use the standard values from fusion physics, so the "too thin or too leaky and it fails" behaviour is the real constraint, not a dramatisation.

One honest caveat: the fusion reactivity used to compute the probability is a good approximation in the 10–20 keV range the sliders cover, but it overstates the rate at very low temperatures, where real fusion collapses much faster due to quantum tunnelling probabilities. So treat the low-temperature percentages as illustrative rather than literal — fine for the concept, worth a word if a sharp student asks.

Using it in class

• Open with the paradox. Ask how splitting and joining can both release energy before you show the curve. Let them sit with the puzzle, then resolve it.
• Run both reactions, then reveal the curve. Do fission, do fusion, note that both release energy — then bring up the binding energy curve as the "aha," with iron at the peak explaining both.
• Make fusion fail first. On the fusion tab, start with the temperature too low so the nuclei bounce apart. The failure is the lesson — it shows the Coulomb barrier is real before you show it being overcome.
• Turn the Lawson dials one at a time. Get a plasma hot enough to fuse, then starve it of density or confinement and watch the burn die anyway. "All three at once" is the insight, and you can only feel it by breaking each condition in turn.
• Compare energy per nucleon, not per reaction. Fission's 173 MeV looks bigger than fusion's 17.6 MeV — until you divide by the number of nucleons involved. That division is where fusion's real advantage shows up.

It sits alongside the rest of the AS91172 suite, so the uranium nucleus splitting here is the same one that jiggles in the nucleus simulator and drives the reactor, and the deuterium–tritium fusion is the same reaction that powers the second stage of a hydrogen bomb. Start with a puzzle that seems to break the rules, and let one curve put it right.

The Dancing, Living, Jiggly Nucleus NCEA 2 and 3 Physics

 

A nucleus that won't hold still: the living U-235 simulator for NCEA Physics

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Draw a nucleus on the board and you'll draw a neat little cluster of circles, sitting still. Every textbook does it. And it plants a quiet misconception that trips students up later: that a nucleus is a static object, a tidy ball of protons and neutrons at rest.

It isn't. A real nucleus is a churning, seething thing — its protons and neutrons in constant, violent motion, held together against forces that are actively trying to tear it apart. This simulator shows exactly that: all 235 nucleons of a uranium-235 nucleus, jiggling in real time, with the tug-of-war that binds them made visible. And crucially, it lets you weaken that binding and watch the nucleus fly apart — which is the fastest way I know to make "binding energy" mean something.

It's a single self-contained web page: open it in a browser, put it on the projector, and it runs. Below I'll walk through the physics it shows, then flag some ways to use it in class.

Two forces, permanently at war

Inside a nucleus, two forces are fighting.

The strong nuclear force pulls nucleons together. It's the glue. But it has a peculiar quality that turns out to matter enormously: it's extremely short-range. A nucleon only feels the strong pull of its immediate neighbours — the ones it's practically touching. Reach across the nucleus and the strong force has faded to nothing.

The electrostatic force pushes the protons apart. Every proton carries a positive charge, and like charges repel. Unlike the strong force, this repulsion is long-range: every proton pushes on every other proton, right across the whole nucleus, no matter how far apart they are.

In the simulator, protons are drawn in one colour and neutrons in another, and you can switch on the forces acting on a single proton to see both at once — the green pull of the strong force dragging it inward, the red push of electrostatic repulsion shoving it outward. The two are very nearly balanced, and that near-balance is the whole story of nuclear stability.

Why the nucleus jiggles

Turn the simulator on and the first thing you notice is that nothing sits still. The nucleons are in perpetual motion, jostling and colliding, the whole nucleus wobbling and deforming as it goes.

This isn't decoration — it's real physics. Nucleons genuinely have kinetic energy; they're moving inside the nucleus all the time. And that motion is one of the things the strong force has to contain. It's not enough for the glue to just balance the electrostatic repulsion; it also has to hold onto nucleons that are constantly trying to bounce their way out. A nucleus is less like a stack of marbles and more like a drop of boiling liquid, held together by its own surface tension while everything inside it seethes.

That "liquid drop" picture isn't a loose analogy, either — it's the actual model physicists use for heavy nuclei, and watching the simulated nucleus wobble and stretch is watching that model run.

Binding energy, made visceral

Here's where the simulator earns its keep. There's a slider labelled binding energy, and a button that removes it entirely.

Binding energy is the energy that would be needed to pull all the nucleons completely apart — to overcome the strong force holding them together. It's an abstract quantity when you first meet it, and students tend to nod along without really feeling what it means. So: wind the slider down.

As the binding weakens, the nucleus swells. It deforms, stretches, becomes visibly less stable — this is exactly what a nucleus looks like in the moments before it fissions. Wind it down far enough, or hit "remove binding energy," and the electrostatic repulsion wins outright. The protons, driven by their mutual repulsion, are flung outward first, and the whole nucleus flies apart across the screen.

That's the moment the concept lands. Binding energy isn't a number in a table — it's the thing that was holding the nucleus together, and you've just watched what happens without it. The nucleus needs that energy, continuously, to exist at all.

A tug-of-war meter on screen makes the balance explicit: strong force holding in, electrostatic force tearing out. At full binding the strong force wins and the nucleus holds; as you weaken it, the meter tips, and at some point repulsion takes over.

Why heavy nuclei are the ones that break

This is the Excellence-level payoff, and the simulator sets it up perfectly, because it's specifically a uranium nucleus — one of the heaviest that exists.

Remember the two ranges. The strong force is short-range: each nucleon is only glued to its nearest neighbours, so adding more nucleons doesn't increase the pull on any individual one very much. But electrostatic repulsion is long-range: every proton you add pushes on all the others, across the whole nucleus. So as nuclei get bigger, the disruptive repulsion grows faster than the strong-force glue can keep up.

Uranium sits right at the edge of this. With 92 protons all repelling each other, its binding energy per nucleon has already started to fall from the peak. The nucleus is only just bound — which is exactly why a small disturbance, like absorbing a single neutron, can tip it over the edge into fission. The simulator lets you feel that precariousness directly: the uranium nucleus is one you can pull apart by nudging the binding down a little, precisely because it was barely holding together to begin with.

And that connects straight to E = mc². The tool shows uranium's numbers — about 7.6 MeV of binding energy per nucleon, roughly 1,800 MeV holding the whole nucleus together, and a mass defect of about 1.9 atomic mass units. That mass defect is the same idea in a different guise: the assembled nucleus is lighter than the separate nucleons it's made from, and the missing mass is the binding energy, via E = mc². Watching the nucleus struggle to hold itself together is watching where that mass went.

A note on the physics

The nucleus in this tool isn't a scripted animation — it's a genuine force simulation. Each nucleon responds to a short-range attraction (the strong force), a hard core that stops them overlapping, and the electrostatic repulsion between protons, with a constant jiggle standing in for their internal kinetic energy. I tuned it numerically so that at full binding it holds together as a stable, jiggling drop that won't quietly drift apart, and so that weakening the binding produces a smooth progression from stable to deforming to flying apart — the real liquid-drop behaviour, not a canned effect.

Two honest caveats worth having in mind. It's a two-dimensional slice, so you're seeing a cross-section of a three-dimensional object. And the forces are scaled for visual clarity rather than set to real MeV values — the behaviour is faithful (short-range glue versus long-range repulsion, heavy nuclei barely bound), but it's a conceptual model, not a quantitative nuclear code. For AS91172 that's exactly the right level: it builds the physical intuition the standard is after, and leaves the hard numbers to the calculations.

Using it in class

• Open it before you define anything. Just let students watch the nucleus jiggle and ask what they're seeing. The fact that it moves is itself the first correction to the textbook picture.
• Point at one proton. Switch on the force arrows for a single proton and narrate the tug-of-war — green in, red out. One particle makes the balance clearer than the whole cluster does.
• Remove the binding energy live. This is the set-piece. Ask students to predict what happens, then hit the button. The protons flying outward first is a detail worth pausing on — it tells you which force won.
• Wind it down slowly. Instead of removing binding all at once, ease the slider down and watch the nucleus deform before it breaks. That deformed, straining state is what a nucleus looks like just before fission.
• Ask why it's uranium. Once they've seen how easily this nucleus comes apart, connect it to why heavy elements are the unstable ones — the long-range repulsion of 92 protons versus the short-range glue.

It sits alongside the rest of the AS91172 suite, so the nucleus you pull apart here is the same one that splits in the fission simulator and powers the reactor. Start with a nucleus that won't hold still, and binding energy stops being a line in a table and becomes the thing you just watched fail.

Nuclear Weapons sim for NCEA 2 and 3 Physics

 

Less than a gram: a nuclear weapons physics simulator for NCEA Physics

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There's one number in this whole topic that stops a class dead. The bomb that destroyed Hiroshima released its energy by converting less than a gram of matter. Not a gram of uranium — a gram of mass, vanished, turned entirely into energy. Everything else in the physics of nuclear weapons is an unpacking of that single sentence, and this simulator is built to make it land.

A word on scope first, because the subject demands it. This tool covers the physics that AS91172 assesses: how much energy comes from how little mass, why a chain reaction can run away, and why a reactor can't. It deals in mechanism and principle only. There is nothing here about how a device is engineered — no critical-mass figures as design targets, no assembly mechanisms, no specifications. Those aren't in the standard and aren't in the tool. What is here is some of the most striking physics in the whole course.

It's a single self-contained web page — open it in a browser, put it on the projector, done. Below I'll walk through the physics across its four screens, then flag some ways to use it.

E = mc², made concrete

Students meet E = mc² as a famous equation long before they meet it as a calculation. The first screen turns it into arithmetic they can follow.

Pick a device — Little Boy, Trinity, Fat Man, or one of two much larger test devices — and the screen shows the energy released and the mass that had to disappear to produce it. For Hiroshima's roughly 15 kilotonnes (6.28 × 10¹³ joules), you rearrange the equation to find the mass:

Δm = E / c² = 6.28 × 10¹³ / (3.00 × 10⁸)² ≈ 7 × 10⁻⁴ kg

Less than a gram. The animation shows that speck of matter on one side and the energy pouring out on the other, and the contrast is the entire point. A city was destroyed by converting a fraction of a gram of matter completely into energy.

There's a second number here that's easy to conflate with the first, and the tool keeps them apart deliberately. About 0.7 grams of mass disappeared, but about 0.76 kilograms of uranium fissioned. Those aren't the same thing. The uranium nuclei that split are still there afterwards — they've become lighter fragments, and it's the tiny difference in mass between the original nucleus and its fragments, multiplied by c², that comes out as energy. Students who grasp that distinction understand mass defect properly; students who don't tend to think the whole kilogram of uranium turned into energy, which is off by a factor of a thousand.

The chain reaction, and the tyranny of exponential growth

One fission releases energy and a couple of spare neutrons. If those neutrons go on to cause further fissions, and those release more neutrons, you have a chain reaction. The second screen lets you control it with a single number: k, the number of neutrons from each fission that go on to cause another.

Set k below 1 and the reaction dies out — each generation is smaller than the last. Set it to exactly 1 and it holds steady, which is a reactor at constant power. Set it above 1 and the numbers double, and double again, and again. This is the difference between a power station and a weapon, and it's a single slider.

But the real lesson of this screen is what exponential growth actually feels like, and it's genuinely counterintuitive. Run the chain reaction and watch the counter. For a 15-kilotonne yield you need about 2 × 10²⁴ fissions — starting from a single neutron, that's roughly 81 doublings. Here's the part that surprises everyone:

• At generation 70, the energy released so far is about 18 tonnes of TNT. Negligible on the scale of the final result.
• At generation 80, it's about 18 kilotonnes — essentially the whole thing.

The entire, city-destroying yield arrives in the last five doublings. For sixty-something generations it looks like almost nothing is happening, and then everything happens at once. With about 10 nanoseconds between generations, the whole event is over in under a millionth of a second — which is why, once it begins, there is no possibility of intervening. That "almost everything arrives at the very end" behaviour is a property of every exponential process, and it's worth sitting with well beyond nuclear physics: it's the same maths behind compound interest, viral spread, and runaway feedback of any kind.

Three designs, one principle

The third screen compares the two 1945 bomb designs and the later hydrogen bomb — as concept diagrams, showing the physical idea rather than any engineering.

The physics that separates the first two is a single idea: critical mass depends on density. Whether a chain reaction can sustain itself comes down to whether neutrons are more likely to hit another nucleus than to escape from the surface — and that depends on how tightly the nuclei are packed, not just on how much material there is.

• The gun-type design (Little Boy) brings two pieces of uranium together. Each alone is too small to sustain a reaction; combined, the assembly is large enough that neutrons find another nucleus before escaping. Simple, and wasteful — only about 1% of the uranium fissioned before the device blew itself apart.
• The implosion design (Trinity, then Fat Man) instead compresses a single sphere. Squeeze the same material into a smaller volume and its density rises, so neutrons travel less far between nuclei and fewer escape. A mass that was subcritical becomes supercritical without adding any material at all. This is far more efficient — Fat Man fissioned a much larger fraction of a much smaller core.

The hydrogen bomb adds a second stage where light nuclei fuse, triggered by the extreme temperature of a fission first stage. Fusion isn't limited by critical mass the way fission is, which is why yields jumped from kilotonnes to megatonnes. And the fusion reaction itself is the same deuterium–tritium physics students meet in the fusion simulator elsewhere in the suite — a nice moment where two topics connect.

There's also a satisfying bit of physics in why these devices are inefficient at all. Once the material is heated to millions of degrees it expands violently, its density drops, and the chain reaction stops. The reaction destroys the very conditions it needs to continue — which is exactly why the gun-type design wasted 99% of its fuel.

The fourth screen, and a sanity check on the numbers

The last screen is a yield comparison on a logarithmic scale, running from conventional explosives up through the 1945 bombs to the largest device ever tested — which released something like 3,300 times the energy of Hiroshima. A log scale is the only honest way to show that range on one chart, and it's a good incidental lesson in why we use log scales at all.

All the energy figures were checked against the physics before they went in: the mass–energy conversions, the fission counts (200 MeV per fission), the number of doublings required. When the tool tells you a 15-kilotonne yield needs about 81 generations, that's calculated, not asserted — and it's the same calculation a student would do by hand, just run instantly as you change the device.

Using it in class

• Lead with the gram. Open on the E = mc² screen and let the "less than a gram" number sit before you explain anything. It buys attention for the rest of the lesson.
• Separate the two masses explicitly. Mass converted versus fuel fissioned is the single most common misconception in this topic. The readouts show both — make students say which is which.
• Run the chain reaction and watch the counter, not the slider. The point isn't that k > 1 grows; it's how it grows. Pause at generation 70, note how little energy there is, then step to 80. The jump does the teaching.
• Tie fission and fusion together. When you reach the hydrogen bomb's second stage, jump back to the fusion simulator. Same D–T reaction, same binding-energy curve — students see the topic as connected rather than a list.
• Use the log scale deliberately. Ask why the yields can't be shown on a normal axis. It's a genuine maths point wearing a physics costume.

The whole thing scales up for projection — there's a zoom control in the corner, or full-screen the browser — and it sits alongside the reactor, fission/fusion and nucleus simulators, so the same E = mc² that runs the power station runs the weapon. That one equation covers both is, quietly, one of the more remarkable facts in the whole of physics.

Ionisation Sim for NCEA 2 and 3

What actually happens when radiation hits matter: an ionisation simulator for NCEA Physics

https://magical-bienenstitch-61644e.netlify.app/

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Ask a class what radiation does, and you'll get answers about danger, glowing green barrels, and superheroes. Ask what it does to a single atom, and the room goes quiet. That gap — between the word "ionising" and the mechanical picture of an electron being knocked loose — is exactly where AS91172 lives, and it's the gap this simulator is built to close.

It's a single self-contained web page. No install, no login, no plugin: open it in a browser, put it on the projector, and it runs. Below I'll walk through the physics it shows, then flag some ways to use it in class.

The one idea everything hangs on

Ionisation is the removal of an electron from a neutral atom. That's it. What's left behind is a positive ion, the freed electron is negative, and together they're called an ion pair. Every hazard, every detector, every dose calculation in this topic is downstream of that one event happening a very large number of times.

The catch that students miss is that it isn't automatic. An atom holds its electrons with a specific amount of energy — the ionisation energy — and a passing particle can only free an electron if it brings at least that much. Below the threshold, nothing happens. The particle scatters away and the atom is untouched.

The first screen of the simulator makes this a button-press. You pick a target atom, set the incoming energy, and fire. If the energy clears the atom's ionisation energy, an electron is knocked free and you're left with a labelled ion pair. If it doesn't, the particle visibly bounces off and the readout tells you why.

The numbers here are real. Hydrogen's ionisation energy is 13.6 eV; helium's is 24.6 eV; sodium's is only 5.1 eV. That spread is the whole lesson. Set the incoming energy to 10 eV and fire it at sodium, then at helium: sodium ionises, helium doesn't. Same particle, same energy, different outcome — because the atoms hold their outer electrons with different grips. It's a concrete way into why the periodic table's ionisation energies vary, without hand-waving.

There's also a live conversion in the working strip, because the standard exam move is to switch between electronvolts and joules:

E = 13.6 eV × 1.60 × 10⁻¹⁹ = 2.18 × 10⁻¹⁸ J

Nothing exotic, but seeing it recompute as you change the target removes the mystery from where the joules come from.

Why alpha, beta and gamma behave so differently

Here's the part that usually gets memorised rather than understood. Students learn that alpha is stopped by paper, beta by aluminium, and gamma needs lead — and they learn it as three unrelated facts. The second screen shows that all three follow from one thing: charge.

An alpha particle carries a charge of +2 and is heavy and slow. Its electric field reaches out and yanks on the electrons of every atom it drifts past, so it creates ion pairs prolifically — several thousand per millimetre in air. But there's a price: it's spending its energy at a furious rate, so it runs out quickly. A few centimetres of air, and it's done. In tissue it manages about 40 micrometres — the thickness of a few cells.

A beta particle is a single electron: charge −1, and far lighter. It moves fast and interacts more weakly with each atom it passes, producing only around ten ion pairs per millimetre in air. Spending its energy slowly, it travels much further — metres of air, or a couple of millimetres of aluminium. In the simulator its track visibly zigzags, because a light particle scatters easily off the atoms it meets.

A gamma ray has no charge at all. This changes everything. With no electric field, it can't reach out and pull on electrons — it only does anything when it happens to score a direct hit on an atom. Those hits are rare, so gamma is the least ionising of the three, and it penetrates the furthest.

When you fire each type into a material in the simulator, you see these differences directly: the dense, short alpha track that stops dead; the sparse, wandering, far-reaching beta; and gamma, which behaves in a completely different way from the other two.

The inverse relationship, and why gamma never truly stops

Put those three behaviours side by side and a pattern jumps out: the most ionising radiation is the least penetrating. Alpha ionises intensely and stops almost immediately; gamma ionises weakly and travels a long way. The third screen makes this explicit — a barrier diagram over a chart of ion pairs per millimetre — but the reason is worth stating plainly, because it's the Excellence-level insight:

Both properties come from the same cause. A particle travels only until it has given away all its energy. Ionise intensely and you spend that energy over a short distance, so you can't penetrate. Ionise weakly and your energy lasts, so you go far. Ionising power and penetrating power aren't two facts — they're two views of one quantity: how much energy the particle transfers per millimetre.

This also dissolves an apparent paradox in radiation safety. Alpha is the least dangerous source outside the body — the dead outer layer of your skin, about 0.07 mm, stops it completely. But it's the most dangerous if it gets inside you, through breathing or swallowing, because in there is no dead layer, and all that intense ionisation is dumped into a tiny volume of living tissue. Gamma is the mirror image: a hazard from outside because it penetrates, but much of it sails straight through you without interacting at all.

One last distinction the simulator is careful about. Alpha and beta have a definite range — a distance at which they simply stop, because they lose energy steadily. Gamma doesn't. It follows exponential attenuation:

I = I₀ e^(−μx)

Each equal thickness of shielding removes the same fraction of the beam, not the same amount. So you can reduce the intensity as much as you like, but it never becomes exactly zero. That's why gamma shielding is quoted as a half-value layer — the thickness that halves the intensity — rather than a "stopping" thickness. Fire gamma into lead and watch the intensity step down: 100%, 50%, 25%, 12.5%… always halving, never quite arriving at nothing. For a class that has just watched alpha stop dead in paper, seeing gamma refuse to stop is the moment the difference becomes real.

A note on the numbers, and one honest caveat

Every figure in the tool was checked against standard references before it went in: alpha's 3.6 cm range in air, its 40-micrometre range in tissue, beta's few metres in air, gamma's 8.6 mm half-value layer in lead. The ranges come from the same empirical formulas used in radiation physics (Geiger's rule for alpha, Katz–Penfold for beta, tabulated attenuation coefficients for gamma), so the trends you demonstrate are the real ones.

Two deliberate honesty choices are worth knowing about. First, the beta energy slider is capped at 3 MeV, because the range formula behind it is only valid up to there — and real beta sources sit below that anyway. Second, the "ion pairs per millimetre" figure is only shown for air, paper and tissue. In a metal it's switched off and marked not applicable, because the ~34 eV-per-ion-pair value is an air/tissue number, and in a metal the electrons are shared across the whole lattice rather than belonging to individual atoms. Metals appear in the tool purely as shielding, judged by how far the radiation gets. It felt more useful to show students where a model stops applying than to print a confident-looking number that isn't real.

Using it in class

A few entry points that have worked well:

• Open on one atom, not on the three radiations. Establish what ionisation is — one electron, one threshold — before touching alpha, beta and gamma. The sodium-versus-helium comparison at a fixed 10 eV is a strong opener.
• Let them predict, then fire. Before running beta into aluminium, ask how far it'll get compared to alpha. The gap between their guess and the result is where the learning is.
• Chase the paradox. Once they've seen alpha stop in paper, ask which radiation they'd least want to eat. The answer surprises them, and the "least penetrating, most ionising" logic clicks.
• Make gamma refuse to stop. After alpha and beta both halt at a definite range, firing gamma into lead and watching it only ever halve is the cleanest possible way into exponential attenuation and the half-value layer.
• Read the working strip aloud. The eV-to-joule conversion and the ion-pairs = E/W calculation update live, laid out the way markers expect, so it doubles as a worked example rather than just an animation.

The whole thing scales up for projection — there's a zoom control in the corner, or just full-screen the browser — and it sits alongside the rest of the AS91172 suite, so the atom being ionised here is the same picture students meet when they get to decay, detection and dose.

If radiation has always been a set of facts your students recite rather than a mechanism they picture, this is an attempt to flip that: start with one electron leaving one atom, and let everything else — penetration, shielding, safety — fall out of it.

Crooke's Electron Tube for NCEA 3 Modern Physics

 This is a sim of Crooke's famous 3 demonstrations from circa 1879 that turned a mysterious glow into the first known subatomic particle--the electron. 

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Photo-electric Effect Simulator for NCEA 3 Physics

 This sim is for NCEA 3 AS91525 "Demonstrate Understanding of Modern Physics". 

Again, the scale is WAAAAY off and this is just to highlight the process. 

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Rutherford's Gold Foil Experiment Simulation

 This is a sim developed for NCEA 2 Physics AS 91172 "Demonstrate understanding of atomic and nuclear physics".

Note: this sim is for illustration purposes only! It is WAAAAAAY out of scale and vastly over-represents the proportion of alpha particles that were deflected or bounced back! 

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Level 2 Physics AS91172: "Demonstrate Understanding of Atomic and Nuclear Physics"

 Here's the series of videos I've done for nuclear physics.

The first one deals with Rutherford's famous gold foil experiment and models of the atom that you'll need to identify and explain in your assessment. 

Part 2 deals with radiation, radioactive decay types that you need to be able to explain (alpha, beta+, beta- and theta decay) and half-lives. 

In part 3 we discuss nuclear fusion, fission and nuclear power. 

 

Circle Geometry NCEA 1.6 The Bare Minimum You Need to Know

 

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Interactive Circle Geometry Tutorial for Section 1.6

This interactive HTML page provides a comprehensive exploration of key circle geometry theorems relevant to curriculum section 1.6. Each theorem is presented in a separate applet with sliders for manipulation, enabling users to observe relationships dynamically. Detailed explanations, proofs, and prompts for deep reflection are included to foster a thorough understanding. Interact with the sliders to vary parameters and consider the guiding questions to enhance conceptual insight.

Theorem 1: Angle at the Center Equals Twice the Angle at the Circumference on the Same Arc

This theorem states that the angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the remaining circumference. This relationship arises from the properties of isosceles triangles formed by radii and the arc.

Proof: Consider a circle with center O and arc AB. Let C be a point on the circumference. Triangles OAC and OBC are isosceles with OA = OC and OB = OC (radii). The base angles are equal. The angle at O is the sum of angles in these triangles, which doubles the angle at C due to the exterior angle property in the formed triangles.

Deep Thinking Prompt: As you adjust the slider to change the arc length, observe how the central angle consistently remains twice the inscribed angle. Why does this hold even for minor or major arcs? What happens as the arc approaches 180 degrees?

Adjust Arc Angle (0° to 180°):

Theorem 2: Angles in the Same Segment (Same Arc) Are Equal

Angles subtended by the same arc at different points on the circumference within the same segment are equal. This follows from the previous theorem, as each inscribed angle is half the central angle for the arc.

Proof: For arc AB, angles at C and D on the circumference are both half the angle at O, hence equal.

Deep Thinking Prompt: Move the slider to reposition one point. Why do the angles remain equal despite different positions? Consider what defines the 'same segment' and how this relates to arc measure.

Adjust Position of Point D (0° to 180°):

Theorem 3: Cyclic Quadrilateral - Opposite Angles Are Supplementary

In a quadrilateral inscribed in a circle (cyclic), the sum of opposite angles is 180°. This is due to the angles being inscribed angles subtending opposite arcs, which sum to 360°.

Proof: Angles A and C subtend arcs BD and AC respectively. The full circle is 360°, so inscribed angles sum to half of that.

Deep Thinking Prompt: Adjust the slider to deform the quadrilateral. Why do opposite angles always sum to 180°? What if the quadrilateral becomes self-intersecting?

Adjust Position of Point D (0° to 360°):

Theorem 4: Angle in a Semicircle Equals 90°

The angle subtended by a diameter at the circumference is always 90°. This is a special case of the inscribed angle theorem where the arc is 180°.

Proof: The central angle is 180°, so the inscribed angle is half, or 90°. Using isosceles triangles, the angles at the ends sum to 90° with the right angle.

Deep Thinking Prompt: Vary the point on the circumference. Why is the angle invariably 90°? Relate this to the Pythagorean theorem in the triangle formed.

Adjust Position on Circumference (0° to 180°):

Theorem 5: Radius Perpendicular to Tangent at the Point of Contact

The radius to the point of contact with a tangent is perpendicular to the tangent line. This ensures the shortest distance from center to tangent is the radius.

Proof: Assume otherwise; a shorter perpendicular would contradict the tangent definition. By Pythagorean theorem in the right triangle formed.

Deep Thinking Prompt: Adjust the tangent position. Why must the angle be 90°? Consider implications for reflection or circle-tangent constructions.

Adjust Point of Contact (0° to 360°):

Theorem 6: Tangent-Chord Angle Equals the Angle in the Opposite Arc

The angle between a tangent and a chord equals the angle subtended by the chord in the alternate segment.

Proof: Using the perpendicular radius and isosceles triangle, angles match via alternate segment properties.

Deep Thinking Prompt: Change the chord position. Why are these angles equal? Explore connections to inscribed angles.

Adjust Chord End Point (0° to 180°):

Theorem 7: Equal Radii Imply Isosceles Triangles; Equal Chords Imply Equal Arcs/Angles

All radii are equal, forming isosceles triangles with the center. Equal chords subtend equal arcs and thus equal central/inscribed angles.

Proof: For isosceles: Two sides are radii. For chords: Congruent triangles with center imply equal arcs.

Deep Thinking Prompt: Adjust chord lengths. Observe equal chords leading to equal arcs. Why do equal arcs imply equal angles? Consider symmetry.

Adjust Second Chord Position (0° to 180°):

```

 

Complete series: 

https://www.youtube.com/playlist?list=PLqWDRCkRvJdwIxIl43BwSkg6Q_IijtS-y

NCEA Right Angle Triangles

 

Complete series: 

https://www.youtube.com/playlist?list=PLqWDRCkRvJdyloP4yNbHkrh0pfdXw3n6F

NCEA 1 Simultaneous Equations

 

The Complete series: 

https://www.youtube.com/playlist?list=PLqWDRCkRvJdyLEyvJWZL5BrARb_lFAEu6

OUR NEXT TOPIC....WITH A VERY SPECIAL GUEST

 

NCEA 1 Parabolas Complete