The usefulness of the k-calculus is its simplicity. It has been successfully used to teach special relativity to young children and also in relativity textbooks.
Many introductions to relativity begin with the concept of velocity and a derivation of the Lorentz transformation. Other concepts such as time dilation, length contraction, the relativity of simultaneity, the resolution of the twins "paradox" and the relativistic Doppler effect are then derived from the Lorentz transformation, all as functions of velocity.
Bondi, in his book Relativity and Common Sense, first published in 1962, reverses the order of presentation. He begins with the Doppler factor, denoted by the letter . From this he explains the twins paradox, and the relativity of simultaneity, time dilation, and length contraction, all in terms of . It is not until later in the exposition that he provides a link between velocity and the Doppler factor . The Lorentz transformation appears towards the end of the book.