  We construct a $C_2$-equivariant spectral sequence for RO$(C_2)$-graded
homotopy groups. The construction is by using the motivic effective slice
filtration and the $C_2$-equivariant Betti realization. We apply the spectral
sequence to compute the RO$(C_2)$-graded homotopy groups of the completed
$C_2$-equivariant connective real $K$-theory spectrum. The computation reproves
the $C_2$-equivariant Adams spectral sequence results by Guillou, Hill, Isaksen
and Ravenel.
