How to use 'exercise' correctly

The word ‘exercise’ is usually wrongly used in the following sentence:
We need to do at least half an hour’s exercise daily to remain healthy.

The above should be corrected as below:
We need to exercise for at least thirty minutes daily to remain healthy.

The usage of ‘exercise’ as a noun refers to the work you have to do for your lessons. For example, you can say ‘I have a lot of exercises to do for Maths today.’

As a verb, ‘exercise’ can also be used in sentences such as:
You can exercise your right as a citizen to vote the leader you like to be a Member of Parliament.