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### Estimating Quotients

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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.
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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/
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Associated Topics:
Elementary Number Sense/About Numbers
Elementary Place Value
Middle School Number Sense/About Numbers

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