Proportion — direct and inverse proportion in algebraic terms

When the relationship involves $d^2$, students use $d$ instead of $d^2$.

Multiplying by $x^2$ instead of dividing by it — the single most common error on this topic.

Writing $y \propto \tfrac{1}{x}$ and stopping — a formula needs an equals sign and a value for $k$.

When the unknown is the variable, not $y$, an extra step is needed. From $F=\dfrac{k}{d^2}$ you reach $d^2=\dfrac{k}{F}$, then must take the square root: $d=\sqrt{k/F}$. Leaving the answer as $d^2$ loses the final mark.

Keep the exact power the statement gives. "$y$ proportional to $\sqrt{x}$" is $y=k\sqrt{x}$, not $y=kx$; "$P$ proportional to the cube of $V$" is $P=kV^3$, not $kV^2$. Changing the power changes every answer that follows.