Math Mistakes that Teachers Hate, Part 1

This is the first in a likely infinite series (math pun intended) of posts on math mistakes that drive me (and all math teachers) crazy!

To kick off this series, I want to start with a mistake that all math teachers of all levels must deal with, from 8th-grade algebra to calculus; a mistake that makes us cringe every time we see it:

Simplify:

If you are a math teacher, you are probably suffering from hypertension and an elevated heartrate right now. If you are a math student, you may be thinking:

If you think the above is correct, then you are causing your math teacher high levels of stress! When I see that mistake, I often feel as though I might turn into the Hulk. I clinch my fist and curse the sky. I cry out in vain ‘Why? Why must this happen to me?’. It is not a pretty sight. Ask my wife.

Regardless of how many times I mention in class that is NOT correct, I still get those type of answers on the homework and exams. Either the student was not in class that day or they were distracted with something else (I’ll talk about cell phones and their societal effects in another post).

If you are math student reading this post, then allow me to educate you. This will save you from the ire of your math teacher. Nothing gets you on a math teacher’s bad side quicker than .

Below is the correct way to simplify . You must simplify this by using the definition of an exponent. An exponent of 2 means to multiply the object it is acting upon (commonly called the base) to itself twice:

Then it is a matter of FOIL (First, Outer, Inner, Last; something most students do quite well):

Observe the following correct example:

Notice that . There is nothing more infuriating and/or disappointing when a student claims that these are equal.

Another (correct) example:

Notice that, again, . Also, . This last one is a mistake that perplexes me even more. Why leave the minus sign there? When you square a real number like -3, the result is positive. Yet, I get the second mistake just as often, if not more so, than the first mistake.

The source of the confusion? As best as I can tell, it is a toss-up between wishful thinking and confusing for . I hope from the bottom of my heart that it is the latter that is the culprit. It is true that:

However, this is just a convenient shortcut for:

If taking a shortcut is going to cause you to get the wrong answer for the much more prevalent problem of , then DON’T TAKE THE SHORTCUT. Always do these problems the same way: apply the definition of exponent and then simplify. If it’s ‘two terms’ times ‘two terms’, then FOIL. If everything is multiplied together, then rearrange the factors and multiply the factors by adding exponents.

If you still insist on using the shortcut, then you must be aware of when the shortcut applies. You can only apply (or distribute) the exponent to FACTORS inside a parentheses. You CANNOT apply an exponent to TERMS inside a parentheses. This is a very important rule.

FYI: Factors are things that are multiplied together. Terms are things that are added/subtracted to one another. It is important that you recognize the difference between a term and a factor. The distinction is important in other areas of algebra and math.

To summarize,

This is okay:

This is NOT okay:

It is important to recognize the difference between and . It may seem like there is only a small difference between the two expressions, but in mathematics, small does not necessarily mean insignificant. Here, it is the difference between having terms and having factors. The shortcut you want to apply only works on factors, not on terms. Hence, this ’small’ difference is quite important. It is the difference between having a shortcut option and not having a shortcut option.

Do your math teacher proud. Stop making the most heart-breaking mistake in all of mathematics. Stop the wishful thinking. Be aware of the differences, however slight they may appear, and what those differences will mean in getting the problem done correctly.

Let me know if you have any questions, comments or suggestions about this post. I hope to do more Math Mistakes that Teachers Hate in the future and I’m sure I could use some suggestions on topics. I will be going on vacation for the next 3 weeks (one of teaching’s greatest perks). So, for any would-be regular readers, I apologize for the long gap in new material that is about to occur. I will post immediately upon my return.

Thanks for reading.

~The Witch Doctor

Tags: algebra, algebra class, algebra teacher, common mistake, help, math, math class, math teacher, mistake, question, teacher