In Fulton's book on toric varieties he (on page 7) goes through the example of $\mathbb{P}^2$.
He considers the fan given by $((1,0),(0,1),(-1,-1))$ and shows via its dual that the variety corresponding to the fan is the gluing of $\operatorname{Spec} k[x,y], \operatorname{Spec} k[x^{-1},x^{-1}y],$ and $\operatorname{Spec} k[y^{-1},xy^{-1}]$.
He concludes his example by saying that if you glue $\operatorname{Spec} k[x,y], \operatorname{Spec} k[x^{-1},x^{-1}y],$ and $\operatorname{Spec} k[y^{-1},xy^{-1}]$ together along the faces, you get $\mathbb{P}^2$. I do not understand this last claim. Perhaps my intuition of the gluing morphism is not good, but I would appreciate any help to improve it.