Estimating Sums and Differences

Date: 10/11/2006 at 01:21:29
From: Janelle
Subject: Estimating Sums and Differences

When estimating sums and differences, I was taught to round each 
number to the greatest place value position of the smallest number.  
Is this the correct method because sometimes I get confused with 
certain numbers.

How would I round 1859 + 997?  Would I round 997 to 1000 and 1859 to
2000, or 997 to 1000 and 1859 to 1900?  Which one is the right way and
why do we do it that way?

Thank you.

Date: 10/11/2006 at 13:14:44
From: Doctor Rick
Subject: Re: Estimating Sums and Differences

Hi, Janelle.

I feel uncomfortable when students (or teachers) talk of a "right way" 
to estimate.  The idea of estimation is to make the work EASY, not to 
get it exactly right!  There is a right way to do the particular kind 
of estimation you are doing right now, but that's because it keeps you 
from doing more work than necessary to get an answer to a certain 
degree of accuracy.

You are told to round the smaller number to one-place accuracy.  The 
first digit in 997 is the hundreds digit, so you want to round it to 
the nearest multiple of 100.  That happens to be 1000 (10 hundreds); 
it may *look* like it's rounded to the nearest thousand, but it isn't!  
The first zero is significant, because the unrounded number is closer 
to 10 hundreds than to 9 (or 11) hundreds.

Now we want to add hundreds.  This means we must round the other 
number to the nearest multiple of 100, also: we round 1859 to 1900. 
Then we can add: 19 hundreds plus 10 hundreds equal 29 hundreds.  Our 
estimate of the sum is 2900.

What would happen if I rounded 1859 to the nearest thousand instead? 
I'd round it to 2000; then I'd add 2000 + 1000 = 3000.  That's an 
estimate, but not as good an estimate!  You can check that the exact 
answer is 1859 + 997 = 2856.  The correct estimate was a better 
estimate than the second estimate: it's off by only 44, compared with 
144.

When you're estimating a sum, you should always round both numbers to 
a multiple of the same number.  If you round one number to the nearest 
multiple of 100 and the other to the nearest multiple of 1000, then 
the hundreds digit of the first number (which is accurate) is added to 
0 hundreds in the second number, and that 0 isn't accurate.  It's a 
waste of effort to add an accurate digit to an inaccurate digit; the 
result will be an inaccurate digit, so you shouldn't pay any attention 
to it.

By the way, if I were told to estimate 1859 + 997 in my head, I 
wouldn't use this method.  In fact, I wouldn't even estimate--there 
is a quick way to get the *exact* answer!  I see that 997 is very 
close to 1000: it is 3 less than 1000.  So I add 1000 to 1859, which 
is easy (2859), then I subtract the 3, to get 2856.  OK, that's a bit 
more work than rounding to the nearest hundred, but not *much* more, 
and I got a *much* better answer!

Right now, you're learning to use a particular tool for estimating. 
Once you have a full toolbox of methods for estimating and for getting 
exact answers in your head, you'll be able to pick whichever tool fits 
best.  Keep that in mind: with the right tools, a worker can do a 
better job--and do it more easily--than someone who has just a few 
basic tools.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/