directed number arithmetic sped up

the rules for combining directed numbers with addition and subtraction are probably too burdensome for memory so some awareness needs to be cultivated, over time

some good graphics have been prepared by Shana McKay here
and dropbox versions here
many thanks to her

once some facility with negative number additions and subtractions has been attained, it is desirable to move towards automating (i.e. no need to think too much) these skills

this is an important step in the mathematics curriculum
many difficulties in algebra can sometimes seem more related to a lack of understanding of directed number techniques 

my view is that sophisticated directed number calculators recognise types of sum and can apply a technique for that general type

presenting students with a sheet of sums, they can work together on grouping the sums by type, explaining that the result (answer) and numbers involved are not important - it's the signs that need to be focused on (the 'form'):

it may be better that students choose how to group these sums themselves, with plenty of discussion,
but another sheet has them already grouped by types:

students can then be given or find some results
and then consider and create general rules for getting results for each type of sum, for themselves

e.g. for type (c) "you subtract the smaller from the larger and give it the sign of the larger (because this 'wins')"

e.g. for type (e) "two minuses don't make a plus; one story could be: if I take £7 from you and then take £12 from you, I've taken £19 from you altogether"

they can also appreciate, after a while, that types can be 'collapsed' to other types
e.g. (e) to (b) and (g) to (a)

and here is a test!