I approached this design with the single-device household in mind. I felt that we should target children, because they tend to be more technologically savvy, and would probably have more time with the device. My intent was to create an interface so that kids could take ownership over the tasks that need to be completed in the house, there is no need to explicitly assign tasks to kids, they can pick them out of the list themselves.
When a task is selected, the child can take a before and after picture, to provide a visual history of the positive changes they make to the house. The child can also see the history of their contributions and their siblings' contributions. The interface still affords the ability to assign a child a task - the parent can take the "before" picture and leave the task incomplete. The partially completed task shows up as a red notification on the child's icon and the task's icon.
For this design I considered the tiny screen scenario, still focusing on the single-device household. The user is presented with a to-do list like interface, checkboxes allow users to mark a task as complete, a plus button allows a user to add a task. Adding and completing tasks open a meta-information panel that allow the adder or completer to add additional information to the task: who completed it, a picture of the task and a picture of the completed task.
For this design I abandoned the single-device household constraint, and imagined a web application that could be opened on multiple computers at once. Information about what tasks are completed is displayed live as they are completed. A user logs into the application beforehand, so the identity context is established (there is a notion of "current user"). A user can create tasks and assign them to himself or other users using a the drag direct-mapping interface. The system notes what user assigned the task and to whom it was assigned. The to-do list is shared across all user's views. If the current user marks a task as completed, he is stating that he completed the task, and the action is broadcasted to all other users of the app.