The Doctor whistles the first few bars of the Pink Floyd song "Another Brick In The Wall" when he sees Clara scolding the students for playing football near the large chess board.
This episode takes place either the next calendar year or the next school year after the events of Into the Dalek (2014). In the earlier story, Danny is introduced as a brand-new teacher at Coal Hill. In this episode, the father of Courtney Woods discusses with Danny how since "last year" his opinion of his daughter has changed, indicating that Danny has been a teacher for some time by this point.
Apart from a couple shots of the Blitzer, every scene in the trailers was taken from the first five minutes of the episode. Except for one scene that, for some reason, is actually taken from Listen (2014).
A banner hangs on the wall saying "A Spirit of Adventure". This is a nod back to Strangers in Space (1964), in which the Doctor tells then-companions Barbara Wright and Ian Chesterton that their travels in the TARDIS with him and Susan Foreman turned out to be "a great spirit of adventure".
The Doctor originally teleported the Blitzer back to its planet of origin, and Danny's interference simply gave it a chance to escape, rather than propelling it three days forward in time.