(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...

recurring decimals and pythagoras

thanks to Mr Toop in Harrogate
and Martin Wilson

somewhat disguised pythagorean triples

maybe avoid using a calculator

ancient Chinese maths in right angled triangles

a ppt gives the problems and suggests approaches to each problem

there are 24 problems on this 'Gougou' topic
the first 10 are here (the rest are included on the ppt)

I have changed problems 7 and 8 to involve more manageable numbers
(my apologies)