geometry steering directed number results

Hans Freudenthal advocates utilising the compelling image of a straight line of integer points when establishing directed number rules for addition and subtraction (e.g. in chapter 15 of his 'Didactical Phenomenology of Mathematical Structures')

in Freudenthal's words:
'the justification of the numerical operations and their laws by the simplicity of the algebraic description of geometrical figures and relations'

a rule e.g. y = 3  -  x generates some points
by inputting some positive x values

• initially less than or equal to 3
• then bigger than 3
• then negative ("where does the line 'say' the result should be?")

as coordinates that have been plotted are joined, the line formed can then be extended into the negative quadrants

some (directed number) calculation results emerge

this work is better managed with talk
but the resources intend to nudge in the desired direction