don steward
mathematics teaching 10 ~ 16

Saturday, 19 March 2011

adding/subtracting directed numbers

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

for addition, a vectorial view seems helpful: start at such and such a place on the number line 'and then'...

for subtraction a 'gap' view/model seems helpful (with due attention to order) especially if, in early stages, the locations are identified on a number line
e.g. 7 - 3 is the gap between (from) 3 and (to) 7, which is 4
e.g. 5 - -2 can be viewed as the gap between (from) -2 and (to) 5, which is 7
e.g. 3 - 8 is the gap between (from) 8 and (to) 3, which is - 5

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 seem more related to a lack of understanding of directed number techniques 

my view is that sophisticated directed number calculators recognise types of sum

presenting students with a sheet of sums, they can work together on the answers and 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 is arguably 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 consider and create general rules 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, if I take £7 from you and then take another £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)

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