I hired a math tutor for my son. Didn't last long, but Iearned a lot. The most useful thing I learned is to say "Explain to me why" after every problem, right or wrong. If he's right, he's checking his work, creating a proof, and teaching you (which happens to be the most efficient way of building long term memory). If he's wrong he will find the mistake.
You don't need to know all that answers, just keep asking "why?"
Since he seems to enjoy mental math, you could check the library for Great Courses Mental Math lecture series. There are some mental math books that break it down step by step and may help you keep up with him.
But, on the other hand, I don't bother trying to keep up with mental math. Mental math is used as a cognitive stress test - it raises blood pressure! I have unstable blood pressure, so mental math is so not my thing! I don't get a headache from higher blood pressure but it sure puts me in a foul, irritable mood!
I think you can get through the summer just with "explain to me why". Another approach could be pushing the level of math until he can't do it in his head anymore. I did that with my son because his little ADHD mind couldn't fathom why a person might need to show work. My son enjoyed AlgebraX on coursera - lots of abstract without many practice problems. I've found a wealth of math instruction DVDs through the library, including many Great Courses.
You don't have to be his "teacher". You can be his resource-finding cheerleader!
