We give a new derivation of the threshold of appearance of the $k$-core of a random graph. Our method uses a hybrid model obtained from a simple model of random graphs based on random functions, and the pairing or configuration model for random graphs with given degree sequence. Our approach also gives a simple derivation of properties of the degree sequence of the $k$-core of a random graph, in particular its relation to multinomial and hence independent Poisson variables. The method is also applied to $d$-uniform hypergraphs.