How this course checks a construction, without a teacher watching
A guessed answer and a built one can look identical sitting still. This is how the difference is found.
A student drags a point to where a perpendicular foot should be. It lands in exactly the right spot. Is that because they constructed it — dropped a genuine perpendicular — or because they have a good eye and got lucky? Sitting still, the two answers are indistinguishable. This article is about the one thing that tells them apart, and why it works.
The idea, stated plainly
A real construction is built from a rule: “wherever A and B end up, C is the point that makes this relationship true.” A guess is built from nothing but a memory of where the answer looked right last time. The two agree at the exact spot where the student first placed things — that is what made the guess look convincing — but they do not have to agree anywhere else.
So: move the givens. If C was really constructed, it has to follow, because the rule that built it still applies to the new positions. If C was guessed, there is no rule connecting it to A and B at all, so nothing pulls it along. It stays exactly where it was left, now visibly wrong.
This course calls that move the wobble test, and it runs it the same way every time: take every free point a student did not build directly, move it to a new position at random, rebuild everything that depends on it, and check whether the thing being tested is still true. Do that several times, from several different random starting points, not just once — a single lucky wobble could still agree with a guess by coincidence, the same way the original guess agreed by coincidence.
Why randomness, and not just one deliberate check
A teacher marking a construction by eye usually drags things around a little and sees if it still looks right. That is the same idea, but a person doing it tends to drag things in ways that feel natural, and “natural” tends to stay close to the shapes they started with. A guessed answer can survive a small, gentle wobble by looking close enough. Reaching for genuinely random positions, several times over, closes that gap: sooner or later the random draw lands somewhere a small deliberate nudge never would have, and that is exactly where a guess runs out of places to hide.
This is not a new idea invented for software. It is the same reasoning behind “the theorem must hold for every triangle, not just the one I drew” — which is why every lesson in this course asks a student to drag the figure themselves before being told the rule, rather than being shown one fixed picture and asked to trust it.
What a three-valued check buys
Sometimes a random wobble produces a degenerate figure by accident — three points that were meant to form a triangle land in a straight line, say, and the question being asked stops making sense for that particular sample. Throwing the whole check out over one unlucky sample would be too strict; but counting a meaningless sample as a pass would be too generous, and counting it as a fail would punish a construction for something that was never really wrong. So each sample’s answer is one of three things — true, false, or “this sample doesn’t apply” — and a sample in that third category is discarded rather than counted either way. The construction still has to pass every sample that actually asked it a real question; it is simply not blamed for the handful of random draws that asked it nothing at all.