This series of tasks are the result of my recent thinking about how algebra emerges from number.  Algebra is the generalistion of number and number relationships.  Why is algebra often taught as something distinct from the number work pupils have already engaged with?

I would suggest that sort of task, in early secondary attempts to bridge the gap between the two.  Perhaps teaching this at the time of number work, but recalling it when later encountering the ideas in algebra.  This is something I am continuing to mull over.  I make no claims to having this “figured out”.  

These tasks are starting points, each one is designed as a prompt to stimulate some mathematical activity.  I think there are a multitude of links here to prime factorisation, index laws etc, etc just waiting to be explored. One idea for a potential part (d) for task 3 might be “create a multi-layered factorisation of 12x^4”. 

Credit: @chrismcgrane84