half an eye
Intermittent or half-hearted attention while something else is happening. Unfortunately, I don't remember what that list said because I had half an eye on the TV while Mom was trying to show it to me.
half an eye
a slight degree of perception or attention. 1962 Cyprian Ekwensi Burning Grass His sandals were new because it was market day; or perhaps he had half an eye to some maiden.