What is the measure of the acute angle formed by the hands of a clock at 4:20 a.m.?  Express your answer as number of degrees.
As the minute hand moves $\frac{1}{3}$ of the way around the clock face from 12 to 4, the hour hand will move $\frac{1}{3}$ of the way from 4 to 5.  So the hour hand will move $\frac{1}{3}$ of $\frac{1}{12}$ of $360^\circ$, or $\boxed{10^\circ}$.