Very good explanation, the smoothie-recipe decomposition approach is the one I used when I taught this to students (I used drinks, but well). However, this example really works only for teaching Fourier Series, which should be the first step to the FT anyway. If you want to understand this stuff, the order you should approach, I think is the Fourier Series, the FT (ignoring many mathematical difficulties), the discrete-time FT, which has its own quirks.