Common Errors
This is a set of errors that really doesn’t fit into any of
the other topics so I included all them here.
Read the instructions!!!!!!
This is probably one of the biggest mistakes that students
make. You’ve got to read the
instructions and the problem statement carefully. Make sure you understand what you are being
asked to do BEFORE you start working the problem
Far too often students run with the assumption : “It’s in
section X so they must want me to ____________.” In many cases you simply can’t assume
that. Do not just skim the instruction
or read the first few words and assume you know the rest.
Instructions will often contain information pertaining to
the steps that your instructor wants to see and the form the final answer must
be in. Also, many math problems can
proceed in several ways depending on one or two words in the problem
statement. If you miss those one or two
words, you may end up going down the wrong path and getting the problem
completely wrong.
Not reading the instructions is probably the biggest source
of point loss for my students.
Pay attention to restrictions on formulas
This is an error that is often compounded by instructors (me
included on occasion, I must admit) that don’t give or make a big deal about
restrictions on formulas. In some cases
the instructors forget the restrictions, in others they seem to have the idea
that the restrictions are so obvious that they don’t need to give them, and in
other cases the instructors just don’t want to be bothered with explaining the
restrictions so they don’t give them.
For instance, in an algebra class you should have run across
the following formula.
The problem is there is a restriction on this formula and
many instructors don’t bother with it and so students aren’t always aware of
it. Even if instructors do give the
restriction on this formula many students forget it as they are rarely faced
with a case where the formula doesn’t work.
Take a look at the following example to see what happens
when the restriction is violated (I’ll give the restriction at the end of
example.)
 This is
certainly a true statement.
 Since and .
 Use the above
property on both roots.
 Since
 Just
a little simplification.
 Since
.
So clearly we’ve got a problem here as we are well aware
that ! The problem arose in step 3. The property that I used has the restriction
that a and b can’t both be negative. It
is okay if one or the other is negative, but they can’t BOTH be negative!
Ignoring this kind of restriction can cause some real
problems as the above example shows.
There is also an example from calculus of this kind of
problem. If you haven’t had calculus
then you can skip this one. One of the
more basic formulas that you’ll get is
This is where most instructors leave it, despite the fact
that there is a fairly important restriction that needs to be given as
well. I suspect most instructors are so
used to using the formula that they just implicitly feel that everyone knows
the restriction and so don’t have to give it.
I know that I’ve done this myself here!
In order to use this formula n MUST be a fixed
constant! In other words you can’t use
the formula to find the derivative of since the exponent is not a fixed
constant. If you tried to use the rule
to find the derivative of you would arrive at
and the correct derivative is,
So, you can see that what we got by incorrectly using the
formula is not even close to the correct answer.
Changing your answer to match the known answer
Since I started writing my own homework problems I don’t run
into this as often as I used to, but it annoyed me so much that I thought I’d
go ahead and include it.
In the past, I’d occasionally assign problems from the text
with answers given in the back. Early in
the semester I would get homework sets that had incorrect work but the correct
answer just blindly copied out of the back.
Rather than go back and find their mistake the students would just copy
the correct answer down in the hope that I’d miss it while grading. While on occasion I’m sure that I did miss it,
when I did catch it, it cost the students far more points than the original
mistake would have cost them.
So, if you do happen to know what the answer is ahead of
time and your answer doesn’t match it GO BACK AND FIND YOUR MISTAKE!!!!! Do not just write the correct answer down and
hope. If you can’t find your mistake
then write down the answer you get, not the known and (hopefully) correct
answer.
I can’t speak for other instructors, but if I see the
correct answer that isn’t supported by your work you will lose far more points
than the original mistake would have cost you had you just written down the
incorrect answer.
Don’t assume you’ll do the work correctly and just
write the answer down
This error is similar to the previous one in that it assumes
that you have the known answer ahead of time.
Occasionally there are problems for which you can get the
answer to intermediate step by looking at the known answer. In these cases do not just assume that your
initial work is correct and write down the intermediate answer from the known
answer without actually doing the work to get the answers to those intermediate
steps.
Do the work and check your answers against the known answer
to make sure you didn’t make a mistake.
If your work doesn’t match the known answer then you know you made a
mistake. Go back and find it.
There are certain problems in a differential equations class
in which if you know the answer ahead of time you can get the roots of a
quadratic equation that you must solve as well as the solution to a system of
equations that you must also solve. I
won’t bore you with the details of these types of problems, but I once had a
student who was notorious for this kind of error.
There was one problem in particular in which he had written
down the quadratic equation and had made a very simple sign mistake, but he
assumed that he would be able to solve the quadratic equation without any
problems so just wrote down the roots of the equation that he got by looking at
the known answer. He then proceeded with
the problem, made a couple more very simple and easy to catch mistakes and
arrived at the system of equations that he needed to solve. Again, because of his mistakes it was the
incorrect system, but he simply assumed he would solve it correctly if he had
done the work and wrote down the answer he got by looking at the solution.
This student received almost no points on this problem
because he decided that in a differential equations class solving a quadratic
equation or a simple system of equations was beneath him and that he would do
it correctly every time if he were to do the work. Therefore, he would skip the work and write
down what he knew the answers to these intermediate steps to be by looking at
the known answer. If he had simply done
the work he would have realized he made a mistake and could have found the
mistakes as they were typically easy to catch mistakes.
So, the moral of the story is DO THE WORK. Don’t just assume that if you were to do the
work you would get the correct answer.
Do the work and if it’s the same as the known answer then you did
everything correctly, if not you made a mistake so go back and find it.
Does your answer make sense?
When you’re done working problems go back and make sure that
your answer makes sense. Often the
problems are such that certain answers just won’t make sense, so once you’ve
gotten an answer ask yourself if it makes sense. If it doesn’t make sense then you’ve probably
made a mistake so go back and try to find it.
Here are a couple of examples that I’ve actually gotten from
students over the years.
In an algebra class we would occasionally work interest
problems where we would invest a certain amount of money in an account that
earned interest at a specific rate for a specific number of year/months/days
depending on the problem. First, if you
are earning interest then the amount of money should grow, so if you end up
with less than you started you’ve made a mistake. Likewise, if you only invest $2000 for a
couple of years at a small interest rate you shouldn’t have a couple of billion
dollars in the account after two years!
Back in my graduate student days I was teaching a trig class
and we were going to try and determine the height of a very well known building
on campus given the length of the shadow and the angle of the sun in the
sky. I doubt that anyone in the class
knew the actual height of the building, but they had to know that it wasn’t
over two miles tall! I actually got an
answer that was over two miles. It clearly
wasn’t a correct answer, but instead of going back to find the mistake (a very
simple mistake was made) the student circled the obviously incorrect answer and
moved on to the next problem.
Often the mistake that gives an obviously incorrect answer
is an easy one to find. So, check your
answer and make sure that they make sense!
Check your work
I can not stress how important this one is! CHECK YOUR WORK! You will often catch simple mistakes by going
back over your work. The best way to do
this, although it’s time consuming, is to put your work away then come back and
rework all the problems and check your new answers to those previously
gotten. This is time consuming and so
can’t always be done, but it is the best way to check your work.
If you don’t have that kind of time available to you, then
at least read through your work. You
won’t catch all the mistakes this way, but you might catch some of the more
glaring mistakes.
Depending on your instructors beliefs about working groups
you might want to check your answer against other students. Some instructors frown on this and want you
to do all your work individually, but if your instructor doesn’t mind this,
it’s a nice way to catch mistakes.
Guilt by association
The title here doesn’t do a good job of describing the kinds
of errors here, but once you see the kind of errors that I’m talking about you
will understand it.
Too often students make the following logic errors. Since the following formula is true
there must be a similar formula for . In
other words, if the formula works for one algebraic operation (i.e. addition,
subtraction, division, and/or multiplication) it must work for all. The problem is that this usually isn’t true! In this case
Likewise, from calculus students make the mistake that
because
the same must be true for a product of functions. Again,
however, it doesn’t work that way!
So, don’t try to extend formulas that work for certain
algebraic operations to all algebraic operations. If you were given a formula for certain
algebraic operation, but not others there was a reason for that. In all likelihood it only works for those
operations in which you were given the formula!
Rounding Errors
For some reason students seem to develop the attitude that
everything must be rounded as much as possible.
This has gone so far that I’ve actually had students who refused to work
with decimals! Every answer was rounded
to the nearest integer, regardless of how wrong that made the answer.
There are simply some problems were rounding too much can
get you in trouble and seriously change the answer. The best example of this is interest problems. Here’s a quick example.
Recall (provided you’ve seen this formula) that if you
invest P dollars at an interest rate
of r that is compounded m times per year, then after t years you will have A dollars where,
So, let’s assume that we invest $10,000 at an interest rate
of 6.5% compounded monthly for 15 years.
So, here’s what we’ve got
Remember that the interest rate is always divided
by 100! So, here’s what we will have
after 15 years.
So, after 15 years we will have $26,442.01. You will notice that I didn’t round until the
very last step and that was only because we were working with money which
usually only has two decimal places.
That is required in these problems.
Here are some examples of rounding to show you how much difference
rounding too much can make. At each step
I’ll round each answer to the give number of decimal places.
First, I’ll do the extreme case of no decimal places at all,
i.e. only integers. This is an extreme
case, but I’ve run across it occasionally.
It’s extreme but it makes the point.
Now, I’ll round to three decimal places.
Now, round to five decimal places.
Finally, round to seven decimal places.
I skipped a couple of possibilities in the
computations. Here is a table of all
possibilities from 0 decimal places to 8.
Decimal places
of rounding

Amount after
15 years

Error in
Answer

0

$10,000.00

$16,442.01 (Under)

1

$10,000.00

$16,442.01 (Under)

2

$60,000.00

$33,557.99 (Over)

3

$24,540.00

$1,902.01 (Under)

4

$26,363.00

$79.01 (Under)

5

$26,457.80

$15.79 (Over)

6

$26,443.59

$1.58 (Over)

7

$26,442.17

$0.16 (Over)

8

$26,442.02

$0.01 (Over)

So, notice that it takes at least 4 digits of rounding to
start getting “close” to the actual answer.
Note as well that in the world of business the answers we got with 4, 5,
6 and 7 decimal places of rounding would probably also be unacceptable. In a few cases (such as banks) where every
penny counts even the last answer would also be unacceptable!
So, the point here is that you must be careful with
rounding. There are some situations
where too much rounding can drastically change the answer!
Bad notation
These are not really errors, but bad notation that always
sets me on edge when I see it. Some
instructors, including me after a while, will take off points for these
things. This is just notational stuff
that you should get out of the habit of writing if you do it. You should reach a certain mathematical
“maturity” after awhile and not use this kind of notation.
First, I see the following all too often,
The just makes no sense! It combines into a negative SO WRITE IT LIKE
THAT! Here’s the correct way,
This is the correct way to do it! I expect my students to do this as well.
Next, one (the number) times something is just the something,
there is no reason to continue to write the one. For instance,
Do not write this as ! The coefficient of one is not needed here
since ! Do not write the coefficient of 1!
This same thing holds for an exponent of one anything to the
first power is the anything so there is usually no reason to write the one
down!
In my classes, I will attempt to stop this behavior with
comments initially, but if that isn’t enough to stop it, I will start taking
points off.