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Front End Estimation

Date: 08/08/2005 at 21:33:50
From: nikki
Subject: I don't know how to do front end estimation

My math class is working on front end estimation, and I just don't get
it.  Can you explain it to me?

Date: 08/08/2005 at 23:13:55
From: Doctor Peterson
Subject: Re: I don't know how to do front end estimation

Hi, Nikki.

I've seen different ways to present front-end estimation (some of
which are much better than others!), so I'm not sure exactly how
you're being taught to do it.  You'd have to check your book or ask
your teacher, or show me what your book says so I can help you
understand it.  But I can tell you the main idea, which I think is
better than following a detailed procedure anyway.

Look at a number like this:


Which digit has the most effect on the value of the number?  The one 
on the left, the 1.  Why?  Because that tells how many hundreds there 
are, while the 2 only tells how many tens you have.  (If you wanted to
quickly count the money in your wallet, it would be more important to
count the hundred dollar bills than the ones, since it would take a
lot of ones to be worth as much as a single hundred.)

So if you want to make a quick estimate of a calculation without
actually doing all the work, you would want to focus on the "front end
digits"--the digits on the left.

As an example, to add the numbers 12345 and 56789, you could ignore
all the digits except the first digit in each, and pretend that the
problem was

  + 50000

That's pretty easy!  You've ignored a lot of stuff, so the answer 
isn't very accurate, but it's a start, and it's certainly a lot closer 
than if you used only the LAST digit in each number and added

  + 00009

Now, you can do a lot to improve the estimate.  Some of these things
you can invent for yourself, just by thinking about how addition (or
whatever operation you are doing) works.  For example, you might look
at the next digit in each number, and think about what would happen if
you included those.  Would the result be different enough from our
simple answer to be worth the trouble?  This sort of thinking can lead
you to use "rounding" to get a better estimate, and also to using the
sizes of the next digits to decide whether to add 1 to the "front-end"
result, or in some cases to round one number up and the other down.

Another direction to think is, what happens if the numbers don't have
the same number of digits?  If we added 12345 and 5678, would we still
want to use the 1 and the 5, or is that not really helpful for making
a good estimate?

One thing to keep in mind when you do estimation problems is that
THERE IS NO REALLY WRONG ANSWER.  Well, there are some really stupid
estimates (like my 14 above), but there is no one RIGHT answer.
Estimation means deliberately getting a wrong answer, when you know
that you don't NEED an exact answer, and you do need a FAST or EASY
answer.  For example, if you are estimating the total of your 
purchases in a store to make sure you don't exceed the amount you have 
in your wallet, you don't need to know the sum right down to the 
penny, but you do need to know how many dollars (or tens of dollars) 
you've spent, and you want to be able to do it in your head.  So the 
kind of thinking we've been talking about will be appropriate there. 
But you might get a slightly different answer than I get, and it 
doesn't matter as long as both of us agree on whether we have enough 

For example, if your estimate is 60,000 and mine is 70,000, when the
actual answer is 69,134, both are reasonable estimates obtained by
slightly different methods.  (The 60,000 is what we got above, and the
70,000 is what I would get by rounding appropriately, which is closer
with hardly any more work.  So mine is better, but yours is still okay.)

So get to it, try it out, and see what you can do, without worrying
about doing it wrong.  I recommend always doing the exact calculation
after making an estimate, then thinking not only about how accurate
your estimate was, but about what you might have done to get closer,
and whether your work was as easy as an estimate ought to be.  You'll
be developing some important skills.

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

- Doctor Peterson, The Math Forum 
Associated Topics:
Elementary Place Value

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