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The Importance of Number Sense and Estimating Answers

Date: 04/28/2004 at 10:32:21
From: Peter
Subject: Using Number Sense

I don't understand how to solve problems using number sense.  For
example, if an employee at the deli counter slices a 2 foot long
salami, about how many slices will he get?  How can I figure that out?
 Why is it important?



Date: 04/28/2004 at 14:35:40
From: Doctor Ian
Subject: Re: Using Number Sense

Hi Peter,

One thing that makes this confusing is that there isn't enough 
information to come up with a definite answer.  That is, the number of
slices he'll get will depend on how thick each slice is.

For example, if each slice is an inch thick, the number of slices must
satisfy this equation:

  24 inches = ? * 1 inch 

We can see pretty easily that there will be 24 slices.  What if each
slice is 1/2 inch thick?  Then we have

  24 inches = ? * 1/2 inch 

So there will be twice as many slices, or 48 slices.  In the general
case, if the width of each slice is W, we have

  24 inches = ? + W

       24/W = ?

Does this make sense?  

So where does number sense come into this?  Basically, you want to 
make a reasonable assumption about the width of a slice, and use that
to get a ballpark figure for how many slices we might end up with.  
I'd probably say that 1/10 of an inch is a reasonable width for a 
slice, and that would give me

   24/(1/10) = 24 * 10 = 240

slices. Maybe there are more, maybe there are fewer, but this is a
_reaonable_ number.  

Why do you care?  Well, suppose someone works the problem and tells
you that the answer is 3, or 150,000.  Would you know right away that
these are ridiculous answers?  The way you'd check them is by making
some reasonable assumptions and using those to come up with an answer
of your own. 

So if someone says "About 500 slices", we could say: "Okay, that's
about twice what we got, so he's making his slices half as thick. 
That's still reasonable."  Or if someone says "About 100 slices", we
could conclude that he's making his slices twice as thick.  Which is
pretty thick, but not unrealistic.  

Again, why do you care?  Well, this is a made-up problem, but as you
get out in the world, you're going to be hearing people throwing all
kinds of numbers around, and you'll need to be able to get some sense
of whether they're realistic.  

For example, a couple of weeks ago I was told by someone soliciting
donations for Mothers Against Drunk Driving that "every 30 seconds, a
person is killed by a drunk driver".  Is that a reasonable number? 
Well, that would be two persons per minute.  There are 60 minutes in
an hour, so that's 120 people per hour.  There are 24 hours in a day,
so that's about 24*120, which is... 

Number sense tells me that 24*120 and 25*120 are pretty close, so I
can use the latter since it's simpler:

  25 * 120 = 25 * 4 * 30

           = 100 * 30

           = 3000

So that's about 3000 people per day.  Again, let's treat a year as 333
days, instead of 365, because it's easier to work with.  (And since we
bumped the number of hours per day up, bumping the number of days down
should balance that out a little.)  So that's 

  3000 * 333 = 1000 * 3 * 333

             = 1000 * 999

             = 999,000

or close to a million people a year.  Which is ridiculously high.  In
fact, if we check the statistics, we find that the actual number is
closer to 20,000 per year.  That's still a lot, but it's 50 times less
than a million!  Which means that the actual figure isn't 30 seconds,
but 50 times more than 30 seconds, or 25 minutes.  

The point is that people throw around all kinds of numbers in efforts
to convince other people to pass laws, raise or lower taxes, donate
money, change lifestyles, declare war, and make other major decisions.
Quite frankly, a lot of these numbers are ridiculous.  But if you 
can't figure out for yourself which ones make sense and which ones
don't, then you just have to believe what you're told, and hope for
the best.  

And cynicism aside, much of what goes on in business and engineering
involves making predictions--e.g., estimating how many burgers people
will probably buy next week, or calculating how much electrical
current will be running through a wire in a given situation.  People
make mistakes (especially where computers or calculators are
involved!) and sometimes they come up with ridiculous predictions. 
Number sense--i.e., being able to quickly come up with ballpark
estimates--can be crucial in detecting these mistakes early on, when
they can still be corrected easily.  

For instance, if you order 10,000 frozen burgers when your freezer
only holds 1,000, you'll end up throwing 9,000 of them in the trash. 
If you use a wire too small for the amount of current that will go
through it, you can burn your house or business down.  It's a lot
easier (and cheaper!) to check your math by making sure your answer is
reasonable.

So, how do you develop this mysterious number sense?  In a word: 
practice.  One good way to get this practice is to make a point of
never using a calculator unless it's absolutely necessary.  When I was
trying to convert "one person every 30 seconds" to an equivalent 
number of people per year, I could have used a calculator, but I
deliberately chose not to, because I knew that whatever number I came
up with would be approximate, instead of exact; and I knew from
experience that I would be able to change the numbers around a little
to end up with an easy calculation that would be "close enough for
government work".  

This might be more than you wanted to hear on the subject!  Can you
let me know if you found it helpful?

- Doctor Ian, 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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