expanding brackets: quadratic expressions

I think it is helpful, wherever possible, for tasks designed to practice and develop a skill to have 'depth'

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 more resiliently


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 

when skills are applied, students can probably better appreciate the benefits of developing particular skills

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

pairs or trios of results in the left-hand column are either identical or closely related - so reasons can be sought for this feature

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 left-hand questions involve a middle term coefficient that is always 10 so students could try to create expressions with this property, maybe making them as complicated as they can

the right-hand questions are the difference of two squares and students could explore and offer a reason why the middle term always disappears

in section (a) the differences between pairs of expressions (questions (1) and (2) etc) is  6
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?