From Longman Dictionary of Contemporary Englishhalf an eye/earhalf an eye/earif you have half an eye on something, or if you are listening with half an ear, you are giving only part of your attention to it He listened with only half an ear and his thoughts wandered. The teacher kept half an eye on them all through the lesson. → half
Examples from the Corpus
half an eye/ear• He told me this and that, but I listened with only half an ear.• Anyone with half an eye could see Susan's antagonism towards her.• With only half an ear for Grigoriev's response, Rostov stared across the room.• She always had half an eye for him; sometimes I thought she watched him as a tamer does a tiger.• Always half an ear, half a mind.• Of course, no government with half an eye on re-election would ever legalise anything it didn't have to.• Allen kept half an eye on the path as he worked.• He has half an eye on where the gun went. 