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4. 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.

\[\sqrt {ab} = \sqrt a \sqrt b \]

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.)

  1. \(\sqrt 1 = \sqrt 1 \)This is certainly a true statement.
  2. \(\sqrt {\left( 1 \right)\left( 1 \right)} = \sqrt {\left( { - 1} \right)\left( { - 1} \right)} \)Because \(1 = \left( 1 \right)\left( 1 \right)\) and \(1 = \left( { - 1} \right)\left( { - 1} \right)\).
  3. \(\sqrt 1 \sqrt 1 = \sqrt { - 1} \sqrt { - 1} \)Use the above property on both roots.
  4. \(\left( 1 \right)\left( 1 \right) = \left( i \right)\left( i \right)\)\(i = \sqrt { - 1} \)
  5. \(1 = {i^2}\)Just a little simplification.
  6. \(1 = -1\)\({i^2} = - 1\)

Clearly we’ve got a problem here as we are well aware that \(1 \ne - 1\)! 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 you can skip this one. One of the more basic formulas that you’ll get is

\[\frac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\]

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 \({x^x}\) since the exponent is not a fixed constant. If you tried to use the rule to find the derivative of \({x^x}\) you would arrive at

\[x\, \cdot \,{x^{x - 1}} = {x^x}\]

and the correct derivative is,

\[\frac{d}{{dx}}\left( {{x^x}} \right) = {x^x}\left( {1 + \ln x} \right)\]

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

\[\sqrt {ab} = \sqrt a \sqrt b \hspace{0.5in} {\mbox{where \(a\) and \(b\) can't both be negative}}\]

there must be a similar formula for \(\sqrt {a + b} \) . 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

\[\sqrt {a + b} \ne \sqrt a + \sqrt b \]

Likewise, from calculus students make the mistake that because

\[{\left( {f + g} \right)^\prime } = f' + g'\]

the same must be true for a product of functions. Again, however, it doesn’t work that way!

\[{\left( {f\,g} \right)^\prime } \ne \left( {f'} \right)\left( {g'} \right)\]

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,

\[A=P{\left( 1+\frac{r}{n} \right)}^{n\,t}\]

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

\[\begin{array}{l}P = 10,000\\r = \frac{{6.5}}{{100}} = 0.065\\n = 12\\t = 15\end{array}\]

Remember that the interest rate is always divided by 100! So, here’s what we will have after 15 years.

\[\begin{eqnarray*}A & = & 10000{\left( {1 + \frac{{0.065}}{{12}}} \right)^{\left( {12} \right)\left( {15} \right)}}\\ & = & 10,000{\left( {{\rm{1}}{\rm{.005416667}}} \right)^{180}}\\ & = & 10,000\left( {{\rm{2}}{\rm{.644200977}}} \right)\\ & = & {\rm{26,442}}{\rm{.00977}}\\ & = & 26,442.01\end{eqnarray*}\]

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.

\[\begin{align*} A & = 10,000{\left( 1+\frac{0.065}{12} \right)}^{\left( 12 \right)\left( 15 \right)} &\\ & = 10,000{\left( 1 \right)}^{180} & 1.005416667=1 \text{ when rounded.} \\ & = 10,000\left( 1 \right) & \\ & = 10,000.00 & \end{align*}\]

It’s extreme but it makes the point.

Now, I’ll round to three decimal places.

\[\begin{align*} A & = 10,000{\left( 1+\frac{0.065}{12} \right)}^{\left( 12 \right)\left( 15 \right)} & \\ & = 10,000{\left( 1.005 \right)}^{180} & 1.005416667=1.005 \text{ when rounded.} \\ & = 10,000\left( 2.454 \right) & 2.454093562 = 2.454 \text{ when rounded.} \\ & = 24,540.00 & \end{align*}\]

Now, round to five decimal places.

\[\begin{align*} A & = 10,000{\left( 1+\frac{0.065}{12} \right)}^{\left( 12 \right)\left( 15 \right)} & \\ & = 10,000{\left( 1.00542 \right)}^{180} & 1.005416667=1.00542 \text{ when rounded.} \\ & = 10,000\left( 2.64578 \right) & 2.645779261=2.64578 \text{ when rounded.} \\ & = 26,457.80 & \end{align*}\]

Finally, round to seven decimal places.

\[\begin{align*} A & = 10,000{\left( 1+\frac{0.065}{12} \right)}^{\left( 12 \right)\left( 15 \right)} & \\ & = 10,000{\left( 1.0054167 \right)}^{180} & 1.005416667=1.0054167 \text{ when rounded.} \\ & = 10,000\left( 2.6442166 \right) & 2.644216599=2.6442166 \text{ when rounded.} \\ & = 26,442.17 & \end{align*}\]

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,

\[2 + x - 6x = 2 + - 5x\]

The \( + - 5\) just makes no sense! It combines into a negative SO WRITE IT LIKE THAT! Here’s the correct way,

\[2 + x - 6x = 2 - 5x\]

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,

\[2 + 7x - 6x = 2 + x\]

Do not write this as \(2 + 1x\) ! The coefficient of one is not needed here since \(1x = x\)! 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!

\[{x^1} = x\]

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.