What's the difference in use between 教 and 教给? At first it seemed like 教给 is just a verb and a preposition, but 教 can be used directly: 我可以教你。
Is that a separate piece of vocabulary?
We use "教" as "teach", e.g., "我可以教你" = "I can teach you", "他教数学" = "He teaches math", "(这道题)我教过你了" = "I have taught you (how to solve) this question."

When we say "教给" we usually mean "give", like "这件事/物品交给你了" = "This thing is given to you，(and from now on you should be responsible for it)", "把这本书教给Tom" = "Give/Bring this book to Tom". However, sometimes we can use "教给" as "teach", depending on "what" you give to someone. For example, "你来教他这个技术" = "(你)把这个技术/方法教给他" = "(You) teach him the technique/way (of doing ...)"