Is it obligatory to buy a tool for the school instead of the one you broke in your youth, or is it sufficient to give charity equal to its value?

Minority and not reaching puberty prevent sin but do not prevent liability. Therefore, you must seek absolution from the school for what you damaged of its equipment. If they do not forgive you, you must pay their due by giving them its value. It is not permissible to give charity with it if it is possible to deliver it and seek absolution directly. The teacher's silence and lack of demand do not affect the establishment of liability upon you.