The bottom number of a time signature can only be part of a sequence starting with 1 and continuing with the double of the number that precedes it. In other words, the numbers of this sequence are: 1, 2, 4, 8, 16, 32, 64, 128… and they continue on in this pattern. The reason for this specific sequence is the fact that each bottom number of a time signature corresponds to a note value (1 = whole note / 2 = half note / 4 = quarter note / 8 = eighth note / 16 = sixteenth note / 32 = thirty-second note …).
One method I use to help my students realize the value of the bottom number of a time signature involves the visualization of fractions. I have them convert the time signature into a fraction. Next, we discard the top number since we are only concerned with the value of the bottom number at the moment. We replace the top number (or numerator) of this imaginary fraction with a 1. The fraction that results from this process represents the name of the rhythmic value that receives one beat in this time signature. A fraction of ½ corresponds to the half note; a fraction of ¼ corresponds to the quarter note and so on.