For some purposes it is convenient to use the definition $$\mathbb{C}P^{\;n-1} = \{ A \in M_n(\mathbb{C}) : A^2 = A^{\dagger} = A, \text{trace}(A) = 1\}$$ This avoids problems if your students are shaky about quotient constructions or confused about considering a line in $\mathbb{C}^n$ as a single point in $\mathbb{C}P^{\;n-1}$. Unfortunately this description is not very compatible with the structure as a complex algebraic variety.