Estimating Quotients

Date: 08/29/2001 at 03:53:46
From: Sherrierobinson
Subject: Estimating Quotients

My daughter is in the 6th grade and is learning how to estimate 
quotients (division). The book I have at home says to round each 
number so that all the digits are zero except the first digit. For 
example, 36,936 would be 40,000. My daughter says that she's taught 
diffently: for instance 36,936 divided by 54 would be 37,000 divided 
by 50; then she is to to cross out the zero in the 50 and one zero in 
the 37,000 (because there was only one zero in the lowest number) and 
her answer would be 740. 

Is she right, or is there something I'm missing? Thanks.

Date: 08/29/2001 at 12:48:19
From: Doctor Peterson
Subject: Re: Estimating Quotients

Hi, Sherrie.

Estimation simply means deliberately making an inaccurate calculation, 
in order to get a reasonably good answer as quickly as possible. That 
means that there is no one "correct" estimation method! Anything you 
do that doesn't take much time and doesn't get too far off from the 
exact answer is a valid estimation. One estimation method may be 
better than another either by being faster or by being more accurate; 
ideally it should be both.

Your book's method rounds both numbers to a single significant digit, 
and will give 40,000/50 = 800, when the exact answer is 36,936/54 = 
684. Your daughter's class's method is to round the dividend to two 
significant digits, giving 37,000/50 = 740. The latter is clearly more 
accurate (because it uses more digits of the dividend), but it's a 
little slower (because it makes you divide more complicated numbers). 
Neither is perfect.

My own method would be something like this: I only want to get a 
single significant digit in the result, for the sake of speed. I want 
the divisor to have a single significant digit, because two-digit 
divisors are too hard to work with in my head; so I round 54 down to 
50. Having decreased the divisor, I know that I should decrease the 
dividend as well in order to keep the quotient close to the exact 
answer. So I want to round 36,936 DOWN to a number whose first two 
digits are an even multiple of 5. That means I round it down to 
35,000. Now I can drop all the zeros and divide 35 by 5, giving 7. I 
dropped three zeros from the dividend and one from the divisor, so I 
have to add back on 3-1=2 zeros to get the final answer:

    36,936 / 54 =~ 35,000 / 50 = 35 / 5 * 100 = 700

And that estimate is not only faster (I think) but also closer than 
either of yours! Both of your methods make the mistake of rounding, 
one up and the other down. 

The essence of estimation is to know numbers well enough to anticipate 
what they are going to do, and take shortcuts that don't get me stuck 
in the woods trying to get to the goal faster. No one set of rules 
will necessarily accomplish this. But students are taught specific 
methods of estimation in order to be able to judge whether they are 
following orders. I'd rather set kids free to figure out a good 
strategy on their own; but there can be some benefit in learning 
several methods, following the one they are told to follow at first, 
and then being allowed to use any method they want in normal practice. 
I don't like your daughter's method much, because it makes you divide 
3700 by 5 without any real benefit from the extra complexity; but if 
it's what she's being taught right now, we can let her use it.

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