"For the better" is an idiomatic expression meaning "to improve" that (usually) functions as an adverb. As with other idioms, you use the whole thing, commonly with a verb that indicates a change:

Hook Street School had not been changed for the better by the war

Her condition took a turn for the better.

The new CEO's cost-cutting measures have been for the better, at least in the opinion of the shareholders.

You do have to be careful to not confuse this with other instances where the three words simply happen to be next to each other, for example:

They'd been walking for the better part of the morning before they reached the first trail marker.

Here "the better part of the morning" is a different idiomatic expression that means "most of the morning".

"For the better" is an idiomatic expression meaning "to improve" that (usually) functions as an adverb. As with other idioms, you use the whole thing, commonly with a verb that indicates a change:

Hook Street School had not been changed for the better by the war

Her condition took a turn for the better.

The new CEO's cost-cutting measures have been for the better, at least in the opinion of the shareholders.

You do have to be careful to not confuse this with other instances where the three words simply happen to be next to each other, for example:

They'd been walking for the better part of the morning before they reached the first trail marker.

Here "the better part of the morning" is a different idiomatic expression that means "most of the morning".