(1)offer  vt（主动）表示愿意（做某事）；提议(英文解释 If you offer to do something, you say that you are willing to do it.）

Peter offered to teach them water-skiing.彼得主动提出教他们滑水。

(2)expect表示指望，期望

You cannot expect to like all the people you will work with.

不要指望你会喜欢所有和你共事的人。

You can't expect to learn a foreign language in a few months.

不要指望在几个月内就能学会一门外语。

根据以上区别，只能填offer

One of the boys helped the boy off the ground and then said sorry to him. He offered  to let him play with them.