Here's how to knit the rib stitch without purl stitches:
Cast on a multiple of 4 + 3 stitches
Row 1 (wrong side): *knit 3, slip 1 with the front yarn*, repeat from * to *, finish with 3 knit stitches
Row 2 (right side): knit 1, *slip 1 with the front yarn, knit 3*. Repeat from * to * until you reach the last 2 stitches, slip 1 with the front yarn, knit 1.
Repeat these two rows to create a rib stitch without purl stitches.