He hurt his back very badly one day and had to give up football. Then he took up swimming to help his back recover.
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What does 'took up' mean here? Would you give me other expressions similar to 'took up'?
In that sentence, "took up" means "began" or "started."
Here are two lists -- phrasal verbs and idioms -- that will help you with various expressions:
http://www.eslcafe.com/pv/ ("take up" is in the T section here)
http://www.eslcafe.com/idioms/id-list.html
In the context you provided, "took up" means to start or begin a new activity or hobby. It suggests that this person started participating in swimming as a way to help their back recover after getting injured.
There are several other expressions similar to "took up" that you can use to convey the same meaning. Here are some examples:
1. Started: "He started swimming to help his back recover."
2. Embarked on: "He embarked on swimming to aid his back's recovery."
3. Engaged in: "He engaged in swimming as a means to heal his back."
4. Adopted: "He adopted swimming to assist in the recovery of his back."
5. Picked up: "He picked up swimming to support his back's healing process."
These expressions all indicate the act of beginning a new activity or hobby in the same way as "took up."