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Explaining Subtraction of Integers as Adding the Opposite

Date: 02/16/2007 at 19:41:41
From: Kristin
Subject: Real-life applications of integer subtraction


I teach grade 7 and we are currently working on how to subtract 
integers.  I like to give students as many examples of real-life 
applications as possible, but I haven't been able to figure out how to 
explain WHY, when subtracting integers, we add the opposite.  Is there 
an explanation or word problem that could help to understand this?

We have been relating integers to money and temperature, but I cannot 
figure out how to do this with a question such as (-5) - (-3).  
Students want the answer to that problem to be -8, not -2.  I have
tried to use the "walking the number line" idea with them, but it
still doesn't answer their question of "When would we encounter this
in real life?"

Date: 02/17/2007 at 13:19:58
From: Doctor Peterson
Subject: Re: Real-life applications of integer subtraction

Hi, Kristin.

Most applications of this are probably hidden in equations; for
example, in solving an equation (as part of solving some real-life
problem) you might subtract x; you never have to worry about whether x
happens to be positive or negative because the same rules apply 
regardless of sign.  This would not be true if we used some different
way of subtracting negative numbers!  That is, the rules are as they
are primarily so that the rules all work and the entire number system
is consistent.  I'll come back to this abstract aspect of the question
in a moment.

There are a couple more visible applications I can think of that might
make this clearer to start with.

The first is using subtraction to find a difference.  How much did the
temperature change if it went from -3 to -5 degrees C?  You find the
answer to this question by subtracting; if the temperatures had been,
say, 73 and 67, we would subtract the old from the new and get 67 - 73
= -6, meaning the temperature went down 6 degrees.  The same happens
in our problem: we subtract the old, -3, from the new, -5, to find
that the temperature went down 2 degrees: -5 - (-3) = -2.  Look at it
on a number line, and it's clear the difference should be 2, not 8.

Why is the answer -2?  The subtraction -5 - (-3) asks what number we
have to add to -3 to get -5; and (-3) + (-2) = -5.  We can find this
-2 by undoing the addition of -3: -5 plus the opposite of -3 is -5 + 3
= -2.

This illustrates the abstract meaning of subtraction: subtracting any
number means undoing an addition (which is the same as adding its
opposite).  That is,

  a - b = c   means   a = b + c

and we can find c by adding -b to both sides of the latter equation:

  a + -b = b + c + -b = b + -b + c = 0 + c = c

So a - b and a + -b are the same thing.

This is the actual proof of this fact on an abstract level, as opposed
to plausibility arguments based on illustrations.

But if you go back to the origins of negative numbers, they were based
not so much on abstract considerations as on money.  Positive numbers
represented assets or gains, while negative numbers represented debts
or losses.  And taking away a debt (reversing a loss, that is,
subtracting a negative number) is equivalent to a gain (adding a
positive number).  For example, if I owe 5 dollars (net), my assets
are -5.  Suppose that, among my accounts, there is a debt of 3
dollars, and a friend forgives the debt, in effect taking away that -3
by paying me 3 dollars.  In subtracting -3 dollars from my account, he
is adding +3 dollars.  Subtracting the negative means the same as
adding a positive.

You can make this idea visible by using play money and IOU slips.  If
I have IOUs for $2 and $3, making a total of -$5 in my pile, then if
someone gives me $3, it will cancel (pay off) the -$3 and leave me
with -$2 in my possession.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 
Associated Topics:
Elementary Subtraction
Middle School Negative Numbers

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