The copy is created on the stack, so it's not "cast." Abilities that trigger when a player casts a spell won't trigger.

After Teach by Example resolves, the next instant or sorcery spell you cast that turn will be copied whether or not it has targets.

If the copied spell divides damage or distributes counters among a number of targets, the division and number of targets can't be changed. If you choose new targets, you must choose the same number of targets.

You can't choose to pay any additional costs for the copy. However, effects based on any additional costs that were paid for the original spell are copied as though those same costs were paid for the copy too.

If the spell has any targets, the copy will have the same targets unless you choose new ones. You may change any number of the targets, including all of them or none of them. The new targets must be legal.

If the spell that's copied has an X whose value was determined as it was cast, the copy will have the same value of X.

If the spell that's copied is modal (that is, it says "Choose one —" or the like), the copy will have the same mode or modes. You can't choose different ones.