Start by going through the expectation values of $S_{z}$, $S_{x}$, and $S_{y}$ for a general time-dependent state with the class. In particular, point out that the expectation of the z-operator has no time dependence while the x and y-operators do.
The third slide has the expectation value of the general spin operator $\vec{S}$, which is composed of the expectation values of the x,y, and z spin operators. Remind students you can't make all of the measurements at the same time, but you can do a large amount of experiments and then combine the expectation values afterwards. This is perfectly fine.
Introduce Ehrenfest's Theorem to the class. Ehrenfest's Theorem states that if you write any classical law for quantum mechanics in terms of expectation values, it will still be true.
This theorem is partially why we commonly see crossover between quantum mechanics and classical mechanics.