Constructing a map of degree 2 $f:T^2\rightarrow S^2$ I know the definition of degree and homology type stuff. But I don't know what a map $T^2\rightarrow S^2$ should actually look like. We never work with explicit examples in my class and I just have no idea what to write. Should I map toroidal coordinates to toroidal coordinates? If not, then what? How do I build this map explicitly? I know his is a stupid question but please help, I'm having trouble with the formalism and I need somebody to directly spell it out for me. Please show me what to do. I don't know how to easily describe maps between spaces like this or how to tell intuitively what I should do to make it have a particular degree. I get what this doing algebraically to the homology groups but I don't know what it is doing to the actual set. 

You want a map $f: S^1  \times S^1 \to S^2$ of degree two. It suffices to provide a map of degree one, since the map $z \times z^2$ is of degree two from the torus to the torus, and degree is functorial.
For a map of degree one, consider the smash product map
$$\gamma: S ^1 \times S^1 \to S^1 \wedge S^1 \cong S^2.$$
To see that this has degree one, you can use the fact that cellular maps are determined by their degrees when collapsing subcomplexes. With the "standard" CW complex structure on $S^1 \times S^1$, $\gamma$ becomes the identity map on the non-collapsed $2$-cell.
