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Using Integers


Date: 01/05/99 at 16:27:02
From: Mary Ellen Hutchind
Subject: Real-life examples of integers

I need real-life examples of integers to show my 7th grade students, to 
demonstrate the fact that they need to be able to use them in math 
class. I've mentioned scores in some games, yardage in football, and 
the balance page of an accountant's book. Can you give me some more? 
Thanks.


Date: 01/05/99 at 17:36:39
From: Doctor Rick
Subject: Re: Real-life examples of integers

Hi, Mary Ellen, welcome to Ask Dr. Math!

I presume you're talking in particular about places where we use 
numbers that may be either positive or negative, and it's really 
helpful to be able to treat them as the same kind of number, instead of 
needing a separate rule for each combination of negative and positive 
numbers. Some examples aren't really integers but positive or negative 
real numbers.

How about temperatures? Conversion between Fahrenheit and Celsius on a
sub-freezing day is a good exercise in using negative numbers.

Elevations go negative in places like Death Valley and the Dead Sea.
Comparing the base-to-peak heights of Mount Everest and Mauna Loa is 
an exercise in subtracting negative numbers.

Latitude and longitude are easier to work with if you take east/north 
as positive and west/south as negative. The most obviously useful 
calculation here is in working with time zones - you'll find charts 
with time zones labeled EST = -5, Eastern Europe = +2, etc.

Years are a peculiar case. AD and BC were invented around AD 525, 
before negative numbers (or zero) were really understood, so there was 
no year zero. The year after 1 BC was AD 1. If it had been done right, 
we would have been able to compute the years between 43 BC and 33 AD 
as 33 - (-43) = 76. But because dates aren't integers, you have to say 
33 + 43 - 1 = 75. In other words, when the calendar was devised, people 
just accepted the need for special cases, but with the invention of 
integers, we found a better way.

By the way, the federal government does not assume that taxpayers 
understand integers, so the 104 form uses the special-case approach: 
"If line 64 is LESS than line 59, subtract line 64 from line 59 and 
write it in line 65. If line 64 is GREATER than line 59, subtract line 
59 from line 64 and write it in line 66." (That's the idea anyway, I 
got my form yesterday but I don't have it in front of me.)

I can't think of any everyday cases where we multiply negatives by
negatives. But when your students learn about quadratic equations, 
they will be benefiting again from the no-special-cases property of 
integers. Early developers of algebra had to present solutions for 6 
kinds of quadratic equations:

  http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Al-Khwarizmi.html

Your students will learn a one-size-fits-all solution method.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
Elementary Number Sense/About Numbers
Middle School Number Sense/About Numbers

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