GCSE content for algebra can be encapsulated into a few main headings:
• algebraic notation and manipulation
• solving equations in one variable and simultaneous equations in two variables
• application of algebra, functions and graphs to real world problems
• properties and transformations of some simple functions.
The roots of these can be found throughout primary mathematics. Working backwards through the curriculum, I can identify eight key strands of algebraic learning and activity that apply, in a limited way, in primary mathematics too:
• Generalising relations between quantities
• Equivalence: different expressions meaning the same thing
• Solving simple equations (finding particular values of variables for particular states)
• Expressing real and mathematical situations algebraically (recognising additive, multiplicative and exponential relations)
• Relating features of graphs to situations (e.g. steepness and direction and turning points)
• Finding new relations from old
• Standard notation
• Understanding functions in general and some specified functions and their properties and behaviour
At least the first seven, and possibly the eighth if graphplotters are used regularly, can be introduced, referred to, and used in a planned and limited way throughout primary school, with a light touch, as the train team might indicate trees on the journey.
I am going to treat these as a 'boxed set' in abbreviated form for the rest of this paper.
• express missing number problems algebraically
• use simple formulae expressed in words
• generate and describe linear number sequences
• find pairs of numbers that satisfy number sentences involving two unknowns.
• enumerate possibilities of combinations of two variables
The accompanying non-statutory guidance makes it clear that it is expected that these should arise from situations that students already understand. Thus using letters is the mathematical way to communicate things that are known.