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Re: neg * neg = pos; why?
Posted:
Jul 23, 1997 2:19 AM
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A rather too long description of one kind of silly but fun approach.
Boy could I have used a picture here!
I've used a number of approaches, such as the two colored chip model
(one side is red, and the other is black), the idea of pumping water
into and siphoning it out of a tank, the postal model of sending and
receiving checks and bills, etc., but my favorite is one I saw years ago
at a math conference, which uses the idea of a hot air balloon. Suppose
it is a calm day, and your hot air balloon is tethered somewhere above
ground at a loading platform labeled 0. If you hook sandbags onto your
balloon, then your balloon will go down, say one unit below the platform
(denoted by negative numbers), for each sand bag. If you hook some
helium bags onto the balloon, it will rise (represented by positive
numbers), say, one unit for each helium bag.
On a transparency, you can draw the number line vertically, and draw on
your platform at 0 and the ladder up to it. Of course you need a nice
little balloon cut out of colored transparency material and some colored
squares (blue for helium and brown or gold for sand bags?) to
demonstrate adding the bags and moving the balloon up and down. So if
you hook on three helium bags and then 5 sand bags, represented by (+3)
+ (-5), you would wind up two units below the platform, or at -2.
The fun comes when you start removing sandbags or air bags. If you take
off helium bags, the balloon goes down. If you take off sand bags, it
will go up. Suppose the balloon is right at the platform, having an
equal number of air bags and sand bags hooked on it. Then suppose you
hook on 3 new helium bags and take off 5 sand bags. Which we could
write as (+3) - (-5). I don't have to tell the students that the
balloon will wind up at +8, or 8 units above the platform.
This leads to some discussion of what we are choosing the signed numbers
to represent and what the operations represent.
Similarly, then, when we get to multiplication, we want to think about
what the numbers and the operation represent. Usually they come up with
this sort of description: The first number will represent the number of
trips up the ladder to adjust the balloon. On each trip we will either
ADD some bags to the balloon, or we will take some bags off the balloon.
This will be represented by the first number in the product. The second
number will describe which kinds of bags we are dealing with on that
trip. So, if you go up the ladder 4 times to ADD 3 helium bags to the
balloon, we would represent that by (+4) x (+3). and you would wind up
12 units above the platform. ((You have a very long hook for doing
this!) If you (starting with the balloon at the platform) make four
trips up the ladder and each time add three sand bags, or (+4) x (-3),
where would you be? And of course, finally, if you start with the
balloon at the platform, all nicely hung with an equal number of helium
bags and sand bags, and make four trips up the ladder, each trip
REMOVING three sand bags on each trip (so (-4) x (-3), sure enough you
wind up 12 units above the platform.
As I've tried to bang this out, I realize how awfully awkward it is to
describe, but with a model, it just takes a few minutes, to act out, a
couple of examples; students get into the act and see what the results
are for themselves almost immediately, everyone seems to have fun, we
talk about this as a very simple example of mathematical modeling - and
how sometimes it may not be a very perfect fit but is good for helping
us develop a sense of what is going on, and how we can stretch our model
only so far to make sense. (I've never been satisfied with trying to
stretch this model to division. You can do it, but it generally seems
to forced to click with the students.) Note that as is often he case
the two factors in the multiplication problem each represent different
things with the resulting question of what the answer represents. (If 4
x 3 represents 4 sacks, with three things in each sack, then I've had
students ask how we decided that the answer describes 12 things rather
than 12 sacks. This, of course leads to the "guzzinta" problem when we
get to division: 3 guzzinta 15 five "times," and few middle or
sometimes even high school students have any idea of what the answer
represents. The answer seems to always be "times" reegardless of what
the original numbers might have represented - but that's another story.
. . . .
Some students have told me that this saved them on things like the GRE
exam when they started to panic in the mathematics section. ("Well, I
just drew my little hot air balloon, calmed right down, and got through
it just fine!") Get out your scissors and have fun - Marj Enneking
Marjorie (Marj) Enneking
Professor of Mathematics
Portland State University
Portland, OR 97207-0751
Phone: 503-725-3643
Fax: 503-725-3661
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