"Per" is used to describe the tool that you use, "kun" is more for accompaniment.
So you might say "Mi iras al la lernejo kun mia amiko" (I am going to the school with my friend) because your friend is just going along with you, but "Mi iras al la lernejo per buso" (I am going to school by bus) because the bus is the tool you use to go there - it is not merely something that happens to be going there with you.
Similarly, "Mi tranĉas la panon kun amiko" (I cut the bread with a friend) indicates accompaniment through "kun", "Mi tranĉas la panon per tranĉilo" (I cut the bread with a knife) indicates the tool you use through "per".
Hard to say without more information.
- What was the prompt?
- What was your response?
- What was the correction?
And most importantly:
- What was the same information for "elsewhere in the same lesson"?
Without the context for the "elsewhere" - there's no way anybody can answer without guessing. Actually, now that I've written all this, I'm starting to think you were asking about "slice" vs "cut."