Translation:The teacher is going to designate a child to do that.
While "ce" can be "this", "that", or "it", "ça" can only be "that" or "it", not "this".
You designate a child. You assign a task. Although you can sometimes assign a person (e.g. to a role), that usage is limited and doesn't fit here. The teacher is giving the child a task (to do), not a role (to be).
In the given context, the use of "assign" here is within the sense of how the action would be understood. "Désigner" can mean to designate, choose, appoint, nominate, or the more mundane "point at", along with some other meanings. So it wasn't at the top of the list, so to speak, but it performs the same job. It will be accepted. Just remember this use so that you can do a reverse translation.