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### Avoiding Careless Mistakes

Date: 01/30/2002 at 03:04:15
From: David
Subject: How to avoid careless mistakes

How to avoid careless mistakes? I have tried do as many problems as
possible, but mistakes are constantly made just because of
carelessness!

I would REALLY appreciate your helping me solve this problem.

Date: 01/30/2002 at 11:10:05
From: Doctor Ian
Subject: Re: how to avoid careless mistakes

Hi David,

I used to have this problem, and in my case the first step toward
getting over it was to read _The Gift of Fire_ and _Less Than Words
Can Say_ by Richard Mitchell.  You can find them online here:

The UnderGrammarian - Richard Mitchell
http://www.sourcetext.com/grammarian/

If you'd like to own them - and I highly recommend this, since it's a
good idea to read them once a year - you can order them from

Self-help guru Anthony Robbins teaches that the hard part about
changing your behavior isn't learning the new behavior, but getting a
very clear understanding of why you really _need_ to change the old
behavior! That kind of understanding is what I got from reading
Mitchell's books. Once I became convinced of the importance of what he
calls 'the habit of correctness and precision', I found that I started
adopting it quite naturally, without much effort at all.

Having said that, it seems to me that many, if not most, of the
careless mistakes that we see here at Ask Dr. Math are caused by
trying to do too many steps at once.  For example,

3(x - 5) = 18

3x - 5 = 18

3x = 23

x = 23/3         Wrong!

Here, the multiplication wasn't fully distributed over the addition in
the first step. This is an easy mistake to make, and the way I avoid
making it is to insert an extra step, in which I leave the parentheses
in place:

3(x - 5) = 18

(3*x - 3*5) = 18

3x - 15 = 18

3x = 33

x = 11         Check:  3(11 - 5) = 3(6) = 18

These 'extra steps' can be especially useful when dealing with
negative numbers, e.g.,

3(x - 5) - 4(x - 6) =

(3*x - 3*5) - (4*x - 4*6) =

(3x - 15) - (4x - 24) =

(3x - 15) + -1(4x - 24) =

(3x - 15) + (-1*4x - -1*24) =

(3x - 15) + (-4x - -24) =

(3x - 15) + (-4x + 24) =

3x - 15 + -4x + 24 =

Now I can be pretty sure that I have the signs correct.  This is more
work than something like

3(x - 5) - 4(x - 6) =

(3x - 15) - (4x - 24) =

3x - 15 - 4x - 24 =

which is much faster, but which also happens to be wrong.  A good rule
to keep in mind that you can't make mistakes fast enough to get a

I also make it a point _always_ to check my answer, after I think I've
found it. At first this is something you have to remember to do, but
after a while it becomes natural.

A second kind of careless error, which sort of falls into the same
category, is caused by translating story problems too quickly into
equations.  For example, here is the kind of thing we see a lot:

The sum of half of a number and 8 less than the number is 25.

(1/2)N - 8 = 25

(1/2)N = 33

N = 66         Wrong!

The mistake can be found with a careful step-by-step translation from
English:

The sum of half of a number and 8 less than the number is 25.
\______/                 \________/

The sum of half of    N     and 8 less than     N      is 25.
\___________/

The sum of      (N/2)       and 8 less than     N      is 25.
\________________/

The sum of      (N/2)       and      (N - 8)           is 25.
\__________________________________________/

(N/2) + (N - 8)                           is 25
\/

(N/2) + (N - 8)                            = 25

(N/2) + (N - 8) = 25

(N/2) + N - 8 = 25

(N/2) + N = 33

(3/2)N = 33

N = (2/3) * 33

N = 22

Check:  The sum of half of 22 and 8 less than 22 is 25.

The sum of 11 and (22 - 8) is 25.

The sum of 11 and 14 is 25.  Yep!

At each step, I identified one small piece of the sentence to be
translated into mathematical symbols. I made the translation, and then
continued as before. If you were doing this on paper, you might do it
more quickly by crossing things out and writing their replacements
above or below, e.g.,

The sum of half of xxxxxxxx and 8 less than xxxxxxxxxx is 25.
N                        N

The sum of xxxxxxxxxxxxxxxx and xxxxxxxxxxxxxxxxxxxxxx is 25.
(1/2)N                   N - 8

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx is 25.
(1/2)N       +           N - 8

and so on. (Here, 'xxxxx' would really look like the original words
with a line drawn through them to show that they no longer need to be
considered.)

On the one hand, working 'in place' is an easy way to get sucked into
making careless mistakes; but on the other hand, the more times you
copy something, the greater the error that you're going to copy it
incorrectly. With a computer, you can simply copy the old line and
change it, which is what I've done here. Without a computer, you have
to use your judgment about whether working in place or copying is more
likely to cause a problem.  To make that judgment, you have to have
some idea about the kinds of mistakes that you tend to make, and how
often you make them.

Which leads to my final recommendation, which is that you might want
to keep a notebook of the careless mistakes you make.  Keeping track
of them would allow you to observe patterns, and figuring out what
you're doing is the first step towards changing _any_ kind of
behavior.

If you find that there are certain _kinds_ of mistakes that you're
making over and over, feel free to write back and tell us what they
are.  That would help us come up with techniques that you could use to
avoid making them in the future.

I hope this helps.

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

Associated Topics:
High School Basic Algebra
High School Negative Numbers