To give up (on something or someone) as hopeless; to throw in the towel or abandon further effort.
彼の不真面目さに先生はついにさじを投げた。
The teacher finally gave up on him because of his lack of seriousness.
あまりに手が回らなくて、彼女はプロジェクトにさじを投げた。
She couldn't handle it anymore and gave up on the project.
何度注意しても直らないので、親もついにさじを投げた。
No matter how many times they warned him he didn't change, so the parents finally gave up.