Computing accurate age and distance factors in cosmology

As the universe expands astronomical observables such as brightness and angular size on the sky change in ways that differ from our simple Cartesian expectation. We show how observed quantities depend on the expansion of space and demonstrate how to calculate such quantities using the Friedmann equations. The general solution to the Friedmann equations requires a numerical solution, which is easily coded in any computing language (including excel). We use these numerical calculations in four projects that help students build their understanding of high-redshift phenomena and cosmology. Instructions for these projects are available as supplementary materials.