The quickest way for converting small numbers like this to binary would probably be to find the closest power of 2, then add/subtract intermediate powers of 2 as needed.
Converting 8 is immediate, since it's already a power of 2:
Converting 15 and 17 is also pretty easy if you're familiar with the rules of arithmetic in binary.
Yes, the student's theory is correct, because the survey shows that the seventh graders spend more time playing games on the weekend than the eighth graders.