Level 7: Precision, scale and dimension

Measurement · level 7 of 8 · rough age 15-16 (a guide for parents; anyone can start anywhere)

Up to now a measurement was a number and a unit. In this level it becomes a number, a unit and an honest statement of how well it is known. You learn how precise an instrument can be, what the digits you write promise, and how errors grow when measurements are combined. Then you treat units as algebra to check formulas and convert anything, see how the square-cube law shapes animals and planets, and finish by estimating things nobody has measured.

Lesson	You will be able to
No measurement is exact	Give the precision and greatest possible error of a reading, write it as \(x \pm \Delta x\) and as an interval, and tell precision from accuracy
Significant figures	Count significant figures, explain why 3.0 m and 3 m differ, and round sums and products honestly
Errors that grow	Combine errors in sums, differences, products, quotients and powers, and find percent error
Dimensional analysis	Check a formula by its dimensions, convert units with factors equal to 1, and find a formula's form from dimensions alone
The square-cube law in nature	Scale lengths, areas and volumes by \(k\), \(k^2\) and \(k^3\), and explain thick elephant legs, cold shrews and slow-falling mice
Fermi estimates	Break an impossible-looking question into guessable pieces, bound unknowns with the geometric mean, and give the order of magnitude

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