The cosine of the laboratory scattering angle is derived for a neutron elastically scattering from a nucleus moving with a specified velocity. Scattering is assumed to be isotropic in the center-of-mass system, and the mean cosine of the laboratory scattering angle is calculated and shown to agree with the first Legendre moment of a scattering probability function derived by Blackshaw and Murray. Isotropic neutron-nucleus encounters are further assumed, and a second average is taken to calculate a mean cosine as a function of the neutron-nuclear speed ratio. This mean cosine approaches 2/(3m), where m is the nucleus mass relative to the neutron mass, as the neutron speed becomes large compared with the speed of the nucleus, but for m > 1, the scattering becomes more anisotropic as this speed ratio decreases before approaching isotropy at small neutron-nucleus speed ratios. This single nuclear speed mean cosine is compared with its average over a Maxwellian distribution of nuclear speeds. The two are qualitatively very similar. Taking the single nuclear speed to be the average speed of the Maxwellian distribution gives better quantitative agreement, in a least-squares sense, between the single-speed mean cosine and the Maxwellian average mean cosine than does using the most probable speed of the Maxwellian distribution.