How is the theme of illusion/appearance versus reality shown in the movie The Matrix?
The Matrix is essentially a spin on the classic philosophical "brain in a vat theory" (read more about that here: http://consc.net/papers/matrix.html). Much like René Descartes's Discourse on the Method and Meditations, the movie explores the dichotomy between mind and matter - or mind and reality. In philosophy, this dichotomy is called Dualism.
The disparity between appearance and reality is brought to the viewer's attention anytime that there is an interaction between the Matrix and something outside of the Matrix. For example, whenever the crewmembers of the Nebuchadnezzar communicate with Tank (whom they call "Operator") while in the Matrix, it reminds the viewer that the Matrix is an illusion. Similarly, the other 'matrices' created by the crewmembers for training purposes are also reminders to the viewer of illusory 'realities,' particularly because these 'matrices' are often altered while the crewmembers are in them.
Countless subtler details point to the disparity between the illusion of the Matrix and the reality that the crewmembers are aware of, such as the 'ports' in the back of their necks or the images of human farms, to name a few.
There are many more examples of the Appearance vs Reality theme in the movie, but I hope these help get you started in your brainstorming.
I should add that I am a college student, not a teacher.