I am trying to use a transitional phrase to link the following paragraph back to the main idea. I am writing about the 3 kinds of schools, private, public and homeschooling. I just wrote about Public now I am transitioning to talk about Private.

My paragraph begins:
However, Private schools require students to wear uniforms and operate differently than public schools.

My question: is this a good transitional phrase? If not how can I correct it?
Thank you,
Jonathon

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[ I am not sure all private schools require uniforms. ]

In contrast to public schools which are free to the families of students private schools require tuition as a rule. Moreover private schools can limit their student body to good students or good football players whereas public schools are open to all.

The transitional phrase you provided, "However," is indeed a good way to transition from discussing public schools to private schools. It indicates a contrast and helps to link the two ideas together. However, if you feel that you would like to explore alternative transitional phrases, you can consider using "On the other hand," "In contrast," or "Conversely" to achieve a similar effect. It ultimately depends on the flow and tone of your overall writing.