While tutoring, I spend a fair amount of time pondering students’ math mistakes. Fortunate for me, then, that I find these mistakes interesting. Believe it or not, I actually collect, categorize and analyze students’ mistakes, for they teach me a lot about students’ struggles with math.
This year, one of the mistakes I’ve been seeing a lot involves one of our more colorful characters in the world of algebra. I’m referring to everyone’s favorite ‘algebraic aunt,’ the relative we all know and love: ‘Dear Aunt Sally.’
As you may recall from your junior high days, ‘Aunt Sally,’ is the lady who guides us in carrying out the order of operations, those steps we use to simplify mathematical expressions. She does so through the cute little phrase that has undoubtedly been passed down since cavemen were doing algebra in the Lascaux caves: “Please Excuse My Dear Aunt Sally” — aka PEMDAS.
You may recall, too (if you haven’t blocked out all the painful memories), that each letter of PEMDAS stands for a different operation: P stands for parentheses, E for Exponents, M for Multiplication, etc.
I’ve never figured out what Aunt Sally ever did that requires us to excuse her over and over, year after year. (Any ideas?) Nevertheless I have discovered something that should qualify for reprehensible behavior by Dear Aunt Sally. It’s the way that the words of her famous expression sow confusion for legions of children.
I’m referring, in particular, to the fact that the “M” of “My” (which stands for “Multiply”) precedes the “D” of “Dear” (which stands for “Divide”). As a result of this unfortunate ordering of letters, many students wind up convinced that — when simplifying mathematical expressions — they ALWAYS perform multiplication before division.
Now, to grasp this next idea, you must understand that usually, while I’m tutoring, students take me at my word. I have a good reputation, and I’ve written a few math books, too. So for the most parts, kids give me plenty of “math cred.”
However, when it comes to “Dear Aunt Sally,” and the fact that I sometimes need to hack away the confusion that sprouts from her phrase like poison ivy from a spring, golly! Do kids get defensive! … Almost as if Aunt Sally is their real aunt, and they need to stand up and defend her …
If I correct the work of a student who has just used this phrase, a more mild child will say: “How can this be wrong? I’m using ‘Aunt Sally!’ ” But the more bold students look at me cannily and say: “I know you’re the tutor, but this time, sorry … you’re just wrong.”
Nevertheless it’s my job to clear up math confusion. So please allow me, the “math ogre” with no abiding love for “Aunt Sally,” to set the record straight.
Just because the “M” of “My” precedes the “D” of “Dear”, that does NOT mean that we ALWAYS multiply before we divide.
The rule actually is this: you do not necessarily perform multiplication before division; nor do you necessarily perform division before multiplication.
So what in the world do you do?
Here’s what: If a mathematical expression contains both multiplication and division symbols, you do WHICHEVER OF THOSE TWO OPERATIONS COMES FIRST AS YOU READ THE EXPRESSION FROM LEFT TO RIGHT.
EXAMPLE: Suppose you’re wrestling with the expression: 12 x 4 ÷ 6. Here, it’s true, you WOULD work out the multiplication before the division. But not because Aunt Sally’s little phrase tells you to do so. No! You do multiplication before division ONLY BECAUSE the multiplication symbol comes before the division symbol as you read the expression from left to right. So this expression gets simplified as follows:
12 x 4 ÷ 6 = (12 x 4) ÷ 6 = 48 ÷ 6 = 8
[Notice that I use parentheses to highlight the operation I’ll perform in the next step.]
But — and this is a big but — if you are working with a slight variation on this expression: 12 ÷ 4 x 6, you would NOT perform the multiplication first. [Haha, take that, Aunt Sally!] Rather, you would perform the division first because the division symbol stands to the left of the multiplication symbol as you read this expression from left to right.
So this expression would be simplified as follows:
12 ÷ 4 x 6 = (12 ÷ 4) x 6 = 3 x 6 = 18
IMPORTANT: Notice that the way you work out an expression can actually change the answer you get. For example, if you simplify the last expression incorrectly, you would get a different answer. This will be wrong (and yes, it’s painful for me to put incorrect math into print), but just to demonstrate the point, I will now do the multiplication before division, like this:
12 ÷ 4 x 6 = 12 ÷ (4 x 6) = 12 ÷ 24 = 12/24 = 1/2 (wrong answer, ouch!)
So the point is that, when performing multiplication and division, you don’t necessarily do the multiplication first. You just do whichever operation appears first as you look at the problem from left to right.
In my next post, I’ll tell you about a similar area of confusion perpetuated by ‘Dear Aunt Sally’ when it comes to addition and subtraction. In the meantime, I suggest you consult your real Uncle Steve or Aunt Suzanna the next time that you need help with math.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two teenage children. Josh is the author of the Parents Choice award-winning Algebra Survival Guide, and its companion Algebra Survival Guide Workbook, both of which will soon be available for homeschoolers as a computer-based Learning Management System, developed and run by Sleek Corp., of Austin, TX.
Josh also authors Turtle Talk, a free monthly newsletter with an engaging “Problem of the Month.” You can subscribe or see a sample issue at http://www.AlgebraWizard.com. Josh also is co-author of the “learn-by-playing” Card Game Roundup books, and author of PreAlgebra Blastoff!, a “Sci-Fi” cartoon math book featuring a playful, hands-on approach to positive and negative numbers.
In the summer Josh leads workshops at homeschooling conferences and tutors homeschoolers nationwide using SKYPE. Contact Josh by email @ josh@SingingTurtle.com or follow him on Facebook, where he poses fun math questions, provides resources and hosts discussions.