A one‐dimensional ice thickness distribution model is presented to determine the minimum number of ice thicknesses necessary to resolve the area‐averaged annual cycles of ice thickness and turbulent fluxes. The baseline case includes 40 ice thickness categories; ice thickness and area, meltwater ponds, ice salinity and age, snow cover, and surface albedo evolve independently for each ice category. A ridging and ice export parameterization, and a coupled one‐dimensional ocean mixed layer model are also included. Sensitivity studies indicate that 16 ice thickness categories can accurately resolve the baseline annual cycles of area‐averaged ice thickness and the summertime turbulent fluxes in this model, provided that one third of the thickness categories represent ice thinner than 0.8 m. Resolving the distribution of ice in this thickness range is important in simulating the area‐averaged ice characteristics. Wintertime values of the turbulent fluxes differ from the baseline by up to 1 W m−2 for fewer than 40 ice thickness categories. The large difference in turbulent fluxes between open water and ice thicker than 1.6 m makes the area‐averaged value sensitive to the number of ice thickness categories, since this number can affect the categories that are merged, the categories that melt completely, ice that is ridged, and the resolution of the ice thickness distribution. This makes an accurate simulation of the baseline wintertime turbulent fluxes difficult. Model simulations with fewer than 40 categories provide a reasonable estimate of the baseline response to a surface longwave heat flux perturbation of greater than 5 W m−2. The annually area‐averaged ice thickness is within 10 cm. The model response to a heat flux perturbation of less than 5 W m−2 is similar for a wide range of ice thickness categories, since the thickness distribution of ice thinner than 1 m is not affected.