Niesen and Chuang, 2nd edition, page 107, Box 2.6, in its motivation for partial trace, says that if M is an observable on system A and $\tilde{M}$ is the corresponding observable on system AB, then it is physically reasonable that the average measurement outcomes computed for the two operators are equal. That is,
$$\mathrm{tr}(M\rho^A)=\mathrm{tr}(\tilde{M}\rho^{AB})=\mathrm{tr}((M\otimes I_B)\rho^{AB})$$
I have no problem with these statements, but they go on to say that this is "obviously satisfied" if $\rho^A=\mathrm{tr}_B(\rho^{AB})$.
This loses me.
The partial trace is defined by $\mathrm{tr}_B(A\otimes B)\equiv A \mathrm{tr}(B)$, I believe.