Negative Numbers in the Real World

Date: Tue, 02 Jul 96 21:02:49
From: Seth Leavitt
Subject: Negative numbers in the real world 

Dear Dr. Math,

I teach 4th grade (5th grade next year) in Minneapolis, Minnesota. 
Math is one of my favorite subjects to teach and as a result, I add 
things that aren't in the curriculum. One of the things I like to talk 
about is negative numbers. 

In elementary math, we try to make things as concrete as possible, 
at least at first. We like to cement the concepts (so to speak) with 
hands-on materials before moving to working with numbers and 
abstractions.

I am finding it difficult to give concrete examples of negative 
numbers. There are negative numbers on a thermometer, but that is 
pretty abstract. Otherwise, I can't think of any good examples.

Do you think the reason for this is because negative numbers are 
not a real concept - that negative numbers are just fictions we agree 
upon in order to manipulate numbers? Or is there such a real thing 
as a negative number in the real (concrete) world? I'd appreciate 
any thoughts you have on this topic. Thank you in advance.

Seth Leavitt

Date: Wed, 3 Jul 1996 18:53:17 -0400 (EDT)
From: Dr. Tom
Subject: Re: Negative numbers in the real world

Hi Seth,

There are plenty of examples.

I'll bet the one that the kids would understand best is negative 
money. If I owe a dollar, I have -1 dollars. In other words, I have 
to get another dollar just to be worthless. If I owe a dollar and 
borrow 3 more, I'm then worth -4 dollars, et cetera. 

Another nice way to think about it is in terms of distances on a 
road. If you live on a road, and measure distances to the west as 
positive numbers and to the left as negative numbers, there's a very 
clean description of position that can be manipulated with positive 
and negative numbers.

For example, if I go west 5 miles, then east 11 miles, then west 2 
miles, starting from my house, I am at: 

0 + 5 - 11 + 2 = -4 -- 4 miles east of the house. 

Elevations work fine, too. Sea level is zero, so if you're in Death 
Valley, 100 feet below sea level, it's easiest to think of your 
elevation as -100 feet. In other words, you have to climb 100 feet 
to get to sea level.

If the inventors of our calendar had understood zero correctly, 
times measured BC and AD would fit nicely into a positive and 
negative numbers. Unfortunately, there is no zero, and as you go 
backward one year at a time, the dates are: 

3AD, 2AD, 1AD, 1BC, 2BC, 3BC.

But you could certainly pick a fixed time, like when you were 
born, and measure all time relative to that -- 2 years before you 
were born is -2; 4 years after you were born is +4. You were born 
at time 0.

Basically, the idea is that if you're measuring stuff that can be 
placed on a line, if the line goes in both directions forever, you 
have to use negative numbers if you want to talk about all possible 
positions. Only rays -- lines that start at a point and go on forever 
from that can be handled with only positive numbers. 

Anything like time, position along a road, temperature, energy, net 
worth, and so on are candidates. Obviously, most roads end, there's 
an absolute zero temperature, and so on, but 4th graders won't 
know that!

-Doctor Tom, The Math Forum