|
|
RE: negative * negative
Posted:
Apr 2, 1996 12:26 PM
|
|
A critical question has been lurking behind a couple questions that
people have raised. Before we talk about multiplying integers, we must
have some understanding of what an integer is. We act as if this is a
trivial, obvious idea, but it is most certainly not! Witness the
historical controversy in an earlier post by my good friend John Owens.
To get really technical, an integer is an ordered pair of whole numbers,
where (a,b) is equivalent to (c,d) iff a + c = b + d. Now I am not
advocating teaching middle school students using set theory, but the point
is that an integer is a RELATIONSHIP between two whole numbers. That's
why they are so difficult. [Side note: Ditto for rational numbers.]
From a mathematical point of view: We generally name an integer using
the representative of its class so that one of the whole numbers is 0,
that is in relationship to zero. So for example, if a>b, then (a,b) =
(a-b,0); which we can consider a positive number. If a<b, then (a,b) =
(0,b-a); a negative. [Remember: the ordered pairs must be whole numbers.]
Thus, when we teach integers on the number line, we are really saying that
an integer is the displacement above or below zero, which is the same as
the displacement between other pairs of whole numbers. So, -3 = (0,3) =
(1,4) = (2,5) ... When we use money, -3 is the change that occurs when we
go from $5 to $2 or from $4 to $1 or from $3 to broke.
When we try to apply operations to integers, things get strange because
we think of operations in terms of whole numbers. Now we are applying
them to RELATIONSHIPS between whole numbers.
Adding is not bad -- join together displacements (as on the number
line). It feels right, much like joining whole numbers (but not the same).
For -3 + -4, we move 3 units to the left, then 4 more units to the left,
for a total of 7 units to the left. [Note: Hitting on the point named -7
is just a shortcut for saying that you ended up 7 units to the left of
where you started from, assuming that you started at 0. If you started at
1, you would end up at -6, which still is the right answer; i.e., a
displacement of 7 units from where you started. An integer is NOT a
location!]
Subtracting is manageable -- it is the inverse of addition, so displace
in the opposite direction. This is analogous to take away with whole
numbers.
The real problem comes with multiplication. There is a trap, because
when we multiply by a whole number, we can think of this as repeated
addition. BUT this is not integer multiplication! This is whole number
multiplication. While the whole number a is related to the integer +a or
(a,0) but IS NOT THE SAME, because an integer is a relationship between
two whole numbers.
Thus, if multiplying by a negative does not seem to flow nicely from
previous ideas of multiplication, there is a reason. We need to add in the
idea of displacement. If we take multiplying by a positive to be like
whole number multiplication, then multiplying by a negative must have the
opposite effect.
SUMMARY: There is a reason that integer multiplication is hard to
explain. Take plenty of time to develop integers so that the students have
a chance to figure out how they work. IT IS NOT OBVIOUS!
PS: This does not mean I know how to do teach this! Just that I believe
it is much harder than we often act. :)
Gary Martin
Univ. of Hawaii
|
|
