In recent years, there has been less of a focus on these sorts of adaptive strategies in secondary maths in Scotland.  This is a problem for 2 reasons.  First, these sorts of tasks are important to support mental strategies.  Having to resort to written algorithms is inefficient if an accurate answer can be obtained mentally.  Secondly, the principles within these questions are the basis upon which formal algebra is taught.  Pupils should, for instance, be comfortable with the idea of distribution long before they “expand brackets” – an action which is generally devoid of meaning for many peoples.  

I am influenced by the work of Tony Gardiner – particularly his task “Finding Hidden Factors” in his book “Extension Mathematics: Alpha” (2007). In question 2, I have created a similar, but slightly different task here.  Questions 2 i, j and k are lifted straight from Tony’s exercise.  

Credit: @chrismcgrane84