Scratch work homework.

Homework that involves scratch work is homework in which you reverse engineer a proof into it's three column scratchwork. Three column scratchwork looks something like this (and the idea is taken from the How to Prove it textbook):

Givens	Goals	Notes
In which you write down the givens, or assumptions.	In which you write down the goals, or what you are trying to prove	In which you write down which proof technique was used to get from this line to the next line. Or, in which you write down the main insights that lead to the application of a given proof technique.

For example, here is the scratchwork for a proof of the theorem if A ⊆ ℘(A) then ℘(A) ⊆ ℘(℘(A)).

Givens	Goals	Notes
A ⊆ ℘(A)	℘(A) ⊆ ℘(℘(A))	Assume that the given is universally quantified. This is the weakest step in the proof.
∀A . A ⊆ ℘(A)	℘(A) ⊆ ℘(℘(A))	Instantiate A in the given with ℘(A)
℘(A) ⊆ ℘(℘(A))	℘(A) ⊆ ℘(℘(A))	Given = goal, so proof is done.