In this paper we provide a construction which produces sequences of pseudo-Anosov mapping classes on surfaces with decreasing Euler characteristic. The construction is based on Penner's examples used in the proof that the minimal dilatation, δ_g,0, for a closed surface of genus g behaves asymptotically like (1/g). We give a bound for the dilatation of the pseudo-Anosov elements of each sequence produced by the construction and use this bound to show that if g_i=rn_i for some rational number r>0 then δ_gi,ni behaves like (1/|ϗ(S_gi,ni)|) where ϗ(S_gi,ni) is the Euler characteristic of the genus g_i surface with n_i punctures.