One of the best aspects of teaching mathematics is watching students think. This is even more enjoyable when students show their own initiative and creativity, overcoming the forced harnesses of rote manipulations so often prescribed. 3/x = 6/(3x-5)
How would you solve for x?
Consider this algebraic statement:
A seventh grader (unfortunately not one of mine) gave the answer "x must be 5" almost immediately after seeing the statement. But how?
He explained: "If you think of going from the left hand fraction to the right hand fraction, the numerator is doubled, so the denominator must be doubled as well. But changing x to 3x means adding one x too man y; this must be exactly compensated by subtracting 5. Thus x must be 5."
Wow....and now as the new school year is in full swing, may you remember this example as you work with students. Don't be too quick in rejecting their problem-solving approaches because they are not the standard or expected approach.
And, take time to listen to a student's reasoning. Speaking from experience, it can be humbling!
Source: Problem and student's explanation is from research done by Robert Davis almost 40 years ago.