I'm working through some algebraic geometry exercises and I stumbled upon the following which I can't seem to get a satisfactory solution to: Prove that there exists a scheme which admits a covering by countably many closed subschemes each of which is isomorphic to $\mathbb{A}_1^k$ (over an algebraically closed field), indexed by $\mathbb{Z}$, such that the copies of $\mathbb{A}^1_k$ corrsponding to i and i+1 intersect in a single point, which is the point 0 when considered as a point in the ith copy, and the point 1 when considered as an element of the (i+1)th copy.
OK, so I have been trying to draw a picture of this and I can see somehow intersecting affine lines to get a geometric figure which satisfies the properties required. However, I feel a bit uneasy with this argument since my construction is very pictorial. So can anyone supply an example , or a hint, of how to construct a scheme that satisfies the requirements above?