warp and weft quadratic expansions

this task uses an idea from John Mason with Alan Graham and Sue Johnston-Wilder in their book 'developing thinking in algebra', published by the Open University in 2005
['going with and across the grain', developed further by Anne Watson and John Mason]

identifying patterns can help to form a method to multiply out bracketed expressions

students are asked to fill in any gaps
and develop ideas about how to continue patterns in rows (right/left) and columns (up/down)
identify what changes and what stays the same as you go across or up/down a grid

they can then complete the array i.e. replace boxes with appropriate expressions

they can also possibly extend the rows and columns further