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Difference between See something with half an eye and With half an eye on something

see something with half an eyesee or understand smth. very clearly or easily:

  • The teacher could see with half an eye that Jim was going to have trouble with some boys who disagreed with him.

with half an eye on something —(also: with one eye on something) directing one’s attention only partly to smth. (while preoccupied with smth. else):

  • I spent the day putting in the web pages with half an eye on the TV.