(a + b)^2 - (a - b)^2

starting with the ancient Babylonians and how some of them based long multiplication on a familiarity with square numbers
moving forward in history
links to the determinant of a quadratic function
links to pythagorean triples

an important diagram?

a ppt is here

a variety of statements can be deduced using the diagram

the 'normal' algebriac method

the function

thoroughly tested...