Chess - I taught Martin the Ruy Lopez which is the most common opening in chess. I did this because this opening is most suitable for beginners, and analyzing the most common moves would be beneficial for Martin. I also taught him en passant, castling, and terms such as skewing, forking and pinning. As well as this, I taught him the optimal positions for pieces (such as knights are more useful in the center of the board).
Tutoring (Josh) - I taught Josh polynomials, such as transformations and how to graph them. Also taught him what degrees of polynomials are and odd and even functions. Josh learned quite quickly and understood most of the concepts.
Tutoring (Tiffany) - I taught Tiffany some trigonometry rules. I learned that she was relying on memorizing the laws of trigonometry and not really understanding it. So I asked her to prove some of the rules that she was memorizing and found out that she was unable to prove most of the trigonometry rules I asked her to prove. So for most of the lesson, I helped her prove the trig rules so she could remember them a lot better. I also taught her some transformations. At the end of the lesson, I gave her some of my work that I did two years ago on trigonometry so she could refer to it when she does the homework I gave her.