As far as my understanding of solid state physics goes, I can't see a clear connection between binding energy and band gap. But our teacher asked us the question as an assignment but i really can't move ahead with anything on this.
Insulators and semiconductors have a band gap. It takes at least this energy to free an electron from a binding valence orbital. In a metal valence electrons are already mobile. To remove an electron altogether from the solid requires an addition energy, the work function. Both quantities are hard to calculate.