ZeroSum Ruler (home)

Blogging on math education and other related things

All too common math mistakes!June 16, 2010

One of my grad school professors called the mistake (a+b)^2 = a^2 + b^2 the “freshman dream”.  I guess he meant freshmen in college, but freshmen in high school make the same mistake.  Come to think of it, a lot of people make the same mistake.  What other mistakes are common in Algebra?

Here are just a few I can think of:

x = 1.     NO.  Well, sometimes, but definitely not always.

-7 + 5 = -12.  NO.  This one’s a never and the reason I developed the ZeroSum ruler.

(a+b)^2 = a^2 + b^2.  NOPE.  There’s a middle term in there.  Write it out.  Don’t be lazy.  Find it.

1/2 = 2.  NO.  “1 out of 2″ is not 2.

(x + 2) + (x - 5) = x^2 – 3x - 10.  NO.  This is an addition problem, not a multiplication problem.

(x+2) – (x - 5) = -3.  NOPE again.  That little – sign is a -1 in disguise and needs to be distributed.

(2/5)(7/3) = ….<blank>….  NO.  Multiply the numerators together and the denominators together to get 14/15.  Easy!

“Thirty percent of 140 = 140/.3″.  NO.  You’re going to get a huge number.  “OF” means to multiply, not divide.

|-5| = 5, so |5| = -5.  NO.  Absolute value is always a positive number.  It represents a distance.  Even if you walk backwards, you’re still moving some distance.

“find x^2 if x = -5″.  Answer: -25.  NO.  Jut because the TI-83 says -25 doesn’t mean it’s right.  Calculators use PEMDAS, which states multiplication comes after exponents.  Remember a few mistakes back that the little – sign is a -1 in disguise.  It’s the same here.  If you put “-5^2″ into the calculator, the calculator will square 5 and then multiply by -1.  To square the entire -5, use ( ).  Or just remember that a negative number raised to any even power will always be positive.

(x^4)(x^5) = x^20.  NO!  This is a common mistake kids make.  But if they start by thinking about what “x^4″and “x^5″ mean, it’s easy to see that (x^4)(x^5) expands to (xxxx)(xxxxx).  Parenthesis right next to each other tell us to multiply, so you end up with (xxxxxxxxx) or x^9 or x^(4+5).

(x^4)^5 = x^9.  No, too.  x^4 = (xxxx) and the “^5″ means you have 5 (xxxx)’s.  So (x^4)^5 expands to (xxxx)(xxxx)(xxxx)(xxxx)(xxxx).  Parenthesis right next to each other still say “multiply” so we have 20 x’s in a row, or x^20.

I know there are more.  What did I miss?