Determine the number of significant figures in a measurement and apply the correct sig-fig rules when adding, subtracting, multiplying, and dividing.
A measurement is only as good as the tool that made it, and significant figures (sig figs) are how chemists report exactly how precise a number is -- no more, no less. The rules: every nonzero digit counts; zeros between nonzero digits count (1002 has 4 sig figs); leading zeros never count (0.0042 has 2 sig figs); trailing zeros count only if there is a decimal point (100 has 1 sig fig, but 100. has 3, and 1.00 x 10^2 has 3).
When you calculate with measurements, the answer cannot pretend to be more precise than your least precise measurement. For adding or subtracting, round the answer to match the fewest decimal places among the numbers used. For multiplying or dividing, round the answer to match the fewest total significant figures among the numbers used.
Scientific notation writes any number as a value between 1 and 10 times a power of 10, and it makes sig figs unambiguous -- every digit shown in the front number is significant. 8,300,000 becomes 8.3 x 10^6 (2 sig figs); 0.00091 becomes 9.1 x 10^-4 (2 sig figs).
Mastery looks like: They correctly count sig figs in any measurement and round calculated answers to the correct precision using the matching add/subtract or multiply/divide rule.
Common stumbles: Applying the multiply/divide rule (count total sig figs) to an addition problem, or vice versa; forgetting that exact counting numbers and defined constants have unlimited sig figs and never limit a calculation.