TL;DR:
Checking by redoing the same steps catches little. Check from a different angle instead.
Run three passes: reread the question, test units and a rough estimate, then check signs and directions.
Most errors that survive are not hard concepts. They are a missed word, a wrong unit or a flipped sign.
You finish the last problem, glance down the page, everything looks fine, and you hand it in. Two days later you get it back with a lost mark for a wrong unit and another for an answer that is obviously too big. Both were findable in thirty seconds.
The trouble with "looking it over" is that your brain reads what it meant to write. A real check forces you to look at the work in a different way than you produced it.
Why rereading your steps finds so little
If you made an error while calculating, you will usually make it again when you repeat the same calculation the same way. A better check tests the answer with a different tool: a unit analysis, a size estimate, a limiting case, or substituting back into the original statement.
What you need
Ten spare minutes at the end. Plan for them before you start.
A different colour pen or pencil, so your checks are visible.
A short list of your own recurring mistakes (signs, units, copying numbers). Add to it every time something is marked wrong.
The three passes
Pass 1: reread the question, then your answer: Read the original question slowly, underlining what is asked. Then read only your final line. Does it answer that sentence, in that form (a speed, a time, a direction)?
Pass 2: units and a rough estimate: Carry the units through the formula. Then ask whether the size is believable using round numbers and everyday sense.
Pass 3: signs and directions: Check every plus and minus, and which way is positive. Look at the sign of the final answer and ask if it should be positive, negative or zero.
Substitute back: Where you can, plug the answer into the original relationship. If it fails, the error is somewhere between the question and the answer.
Worked example: two physics problems, three passes
Take two problems from a problem set. Use g = 9.8 m/s² and ignore air resistance.
Problem A. A block slides without friction down a ramp. It starts from rest at a vertical height of 3.0 m. What is its speed at the bottom? Your first attempt reads: v² = 2gh = 2 × 9.8 × 3.0 = 58.8, so "v = 58.8 m/s".
Pass 1. The question asks for speed, and the line above has
v²on the left. The final line says v = 58.8, which does not follow: you found v², not v.Pass 2. Units of 58.8 are
m²/s², which are not speed units. An estimate agrees: 59 m/s is about 210 km/h, which is far too fast for a 3 m ramp. A 3 m drop is like a 3 m free fall, and an object falling from a 3 m ledge hits the ground at a jog-to-run speed, a few metres per second.Fix.
v = √58.8 ≈ 7.7 m/s. Check:7.7² = 59.3, which is near 58.8, as expected after rounding.
Problem B. A ball is thrown straight up at 12 m/s. How long until it reaches the top, and how high does it go? Your attempt takes up as positive and writes v = v₀ + at with a = +9.8, so 0 = 12 + 9.8t and t = −1.22 s.
Pass 1. The question asks for a time to reach the top. A time cannot be negative here, so something is wrong.
Pass 3. You chose up as positive, so gravity, which points down, must be
−9.8. Redo:0 = 12 − 9.8t, sot = 12 / 9.8 = 1.22 s.Height.
h = v₀t + ½at² = 12 × 1.2245 − ½ × 9.8 × 1.2245² = 14.69 − 7.35 = 7.35 m.Substitute back. A second method:
h = v₀² / (2g) = 144 / 19.6 = 7.35 m. Both give 7.35 m.
Notice what found each error. Neither was a hard idea. Problem A needed a square root and a unit glance. Problem B needed a decision about which way is positive. Each pass took under a minute.
Problem | First answer | What caught it | Corrected |
|---|---|---|---|
A (ramp) | 58.8 m/s | units m²/s², too fast | 7.7 m/s |
B (ball) | t = −1.22 s | negative time, sign of g | t = 1.22 s, h = 7.35 m |
Mistakes to avoid
Checking only the problems you already feel unsure about. Confident answers hide as many slips.
Redoing the arithmetic on a calculator the same way and calling it verified.
Skipping the estimate because it feels too crude. A crude estimate is what catches errors of a factor of ten.
Rounding in the middle and then comparing to a precise value.
Running out of time because you left no minutes for checking.
Your plan for this week
On your next homework set, reserve the last ten minutes for the three passes.
Mark each catch with a different colour and write one word for the type: unit, sign, question, estimate.
At the end of the week count your types. Your most common one is your personal checklist item for the next set.
Frequently asked questions
How do I check when there is no time on a test?
Shrink the routine: reread the question's last sentence, glance at the units, and ask if the size is sensible. That takes about ten seconds per problem.
What if my check disagrees with my answer?
Trust neither yet. Redo the problem a different way and see which one the second method agrees with.
Get started
