(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...

two, 2-digit multiplications

four tasks with a common theme:

1.   (10n + 5) squared
2.   from a square number, go a bit more and a bit less and multiply them
3.   31 squared and 29 squared
4.   units sum to 10, in a long multiplication sum

these involve (a + b) squared, the difference of two squares and (a - b) squared

they could be a starting place for a study of these expansions

all start with number calculations (long multiplication practice) and move on to consider 

related images (an array) and then 

expanding quadratic brackets examples (which play automatically)

ppts are here:

1.  (10n + 5)^2

2.  24 x 16 

3.  31 ^ 2

4.  42 x 28

task 1

task 2

task 3

task 4