The class would probably be best served with a chronological ordering of events. Start with pinball machines, early arcades, colecovision and atari and talk about how the genre starts with nothing but sound effects and single-track audio jingles.
The simultaneous development of "chip" music (music produced by running signals through the hardware itself) on Japan's Famicom and PC-88, and the west's Commodore Amiga (more popular in EU than America) is where looping "background music" (or "BGM") really got its start in gaming.
A great case study for the early hardware would most definitely be Koichi Sugiyama, whose compositions were originally written for a full symphony orchestra, recorded as such, but then "de-made" to fit the limitations of the hardware (for the first 4 DQs, that would be Famicom, then SFC for 4's remake and 5 and 6). Sugiyama didn't think of himself necessarily is a media (film/TV/game) score guru, but as a neo-classical and neo-romantic, and very "traditionalist," composer.
From there, starting with the MIDI sequencing and sound banks of the 16-bit consoles, things get a lot more complicated. But I'd always keep the focus of the course, when contrasting it to film, as the difference between needing catchy, "loopable" (indefinitely playing) music vs very sharp, evocative tunes playing at just the right times (that is, music cues). Eventually as games became more cinematic, we'd have more examples of non-looping "cue" or "cut scene" music. The earliest type of this music I know of was Tecmo's ahead-of-its-time cut scenes for Ninja Gaiden. But obviously, starting with the FMV revolution of the 32-bit era, it starts to become more and more common, til eventually you get to games like Halo and Alan Wake whose music is almost entirely cinematic and non-looping in nature.
Does this help?