this can be achieved by involving some regularity (a pattern) to results or a general form/structure - that could lead on to proof considerations
as Dave Hewitt reminds us, stressing structure helps someone see generality quicker and deeper
it seems helpful for skill development to have some application, either:
- within maths (e.g. to more firmly establish a result) or
- within some practical context
by way of examples of 'tasks with depth' (not necessarily good ones...) here are some tasks involving work on expanding the product of two linear expressions
the right-hand column involves the skill of expanding brackets leading to the justification of a result - which might better be initiated from numerical examples (e.g. questions 18 and 19 involving any four consecutive numbers)
the right-hand questions are the difference of two squares and students could explore and offer a reason why the middle term always disappears
can students create their own examples and possibly give reasons for this 'gap'?
in section (b) the pairs of expressions are closely related
- can students create their own examples and consider some reasons why the expressions are so close?
for these questions, the result should either be a number or
a quadratic expression that factorises
can students create their own, similar, questions?