In Algebra I, we drill into student’s heads the formula for the difference of two squares:
While this formula can be confirmed by just multiplying out the right-hand side, innovative teachers can try to get students to do some exploration to guess the formula for themselves. For example, teachers can use some cleverly chosen multiplication problems:
Students should be able to recognize the pattern (perhaps with a little prompting):
Students should hopefully recognize the perfect squares:
so that they can guess the answer to something like without pulling out their calculators.
Students should be able to recognize the pattern:
and perhaps they can even see the next step:
From this point, it’s a straightforward jump to
leading students to guess that .