Nobody marks your checker but you. Every mathematician you will ever meet runs one — a quiet second pass that catches errors before they matter — and the earlier you build yours, the sooner tests stop being scary. Four habits make up the whole machine.
Interrogate the candidate — into everything
Solved an optimization? The derivative handed you a candidate, not an answer — nudge the variable a little each way and the function itself will referee: at a true maximum, both nudges lose. A claimed derivative can be cross-examined the same way, no answer key required: if you believe , then had better come out a whisker from 6. The definition is a referee you carry everywhere. The same reflex checks a model: feed your braking-car curve a radar reading you did not use to build it, and see whether the position it predicts matches the data.
Estimate first, compare after
Before solving, write down a rough expected size — the reflex Estimation Duels trains. The tangent slope of at must land between 5 and 7, because the secants on either side bracket it; if your algebra hands you 12, one of you is wrong, and now you know to look. This check only works if the estimate came before the answer did.
Check the units — and the species
Units are a free error detector. If a velocity answer arrives in metres when the question asked for metres per second, something upstream broke — and the units caught it without you rereading a single step. In the vector unit the check grows teeth: a dot product is a scalar — if yours came out with components, something broke; a cross product is a vector — if yours came out as a plain number, same alarm. Answers here have a species as well as a size, and saying both out loud is the cheapest check in the course. Carry them through every line, as Showing Your Thinking insists.
Ask whether the graph agrees with its own story
After building a sign chart, interrogate the sketch. A positive derivative promises a climbing curve — if your sketch falls where your chart says positive, one of them is lying, and it is not the chart. A velocity graph that never touches zero belongs to a position graph with no turning point; a descent model whose height dips below the runway is promising a landing that goes through the ground, and the sketch objects instantly. A graph that disagrees with its own situation is the cheapest alarm in Curve Sketching — and the question behind it is the one What Makes a Model Good asks of every model: does it tell the truth about the world it claims to describe?