by Josh Rappaport
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, as long as we are ganging up on Dear Aunt Sally, there's another critical issue we must address. The very thing that we are saying about multiplication and division also applies to addition and subtraction. This is so even though the "A" of "Aunt" (which stands for "Addition") precedes the "S" of "Sally (which stands for "Subtraction.").
This means that if a problem contains both addition and subtraction symbols, you don't necessarily perform the addition before performing the subtraction. Nor vice-versa. Rather, for addition and subtraction, you do whichever operation appears first as you read the expression from left to right.
EXAMPLE: In the problem: 10 – 2 + 6, you subtract 2 before you add 6, as follows:
10 – 2 + 6 = (10 – 2) + 6 = 8 + 6 = 14
Notice that if you incorrectly perform the addition first, you would get a different answer, as follows:
10 – 2 + 6 = 10 – (2 + 6) = 10 – 8 = 2 (incorrect answer)
How to make sense of all of this? Here's how I think of it. There are hierarchies in the math world just as there are hierarchies in the "real world." In the math world, one hierarchy deals with operations, and it states: multiplication and division are on the same level; they have the same power. Since these two operations are on the same level, there's no obvious way to tell which of those operation we should do first. So we will create a way. From here on out (this was said a long time ago!), if we want people to multiply first, we'll set up the expression in such a way that the multiplication symbol stands to the left of the division symbol. And vice-versa: if we want people to divide before multiplying, we'll set it up with the division symbol to the left of the multiplication symbol.
Now, since multiplication and division are on the same level, we can view multiplication and division as forming one hierarchy level, the multiplication/division level.
Similarly, another aspect of the math hierarchy states that addition and subtraction are on the same level. So those two operations form the addition/subtraction level.
However, there is a difference in hierarchy between the addition/subtraction level and the multiplication/division level. The multiplication/division level is higher, more powerful (in a sense), than the addition/subtraction level. That is why, given a problem containing, say, both multiplication and addition, you would always multiply before you add, no matter which term is to the left or to the right. But given a problem with multiplication and division, you need to use the left-right relationship to decide which of these "same-power" operations you do first.
One last question/answer unit. Suppose a problem has all four operations in it. How do you handle all of those operations? Here's what you need to know.
If a problem has all four operations, you perform the multiplication/division operations first. Then, once those have been performed, you perform the addition/subtraction operations. The following example illustrates this idea:
PROBLEM: 24 – 12 ÷ 2 + 3 x 3
24 – 12 ÷ 2 + 3 x 3 [problem]
= 24 – (12 ÷ 2) + 3 x 3 [set off division]
= 24 – 6 + 3 x 3 [do division]
= 24 – 6 + (3 x 3) [set off multiplication]
= 24 – 6 + 9 [do multiplication]
= (24 – 6) + 9 [set off subtraction]
= 18 + 9 [do subtraction]
= 27 [do addition]
Moral of the story: don’t let 'Dear Aunt Sally' fool you into thinking that you must do multiplication before division, or that you must do addition before subtraction. For each hierarchy level — multiplication/division or addition/subtraction — you do whichever operation comes first as you read the expression from left to right.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two teenage children. Josh is the author of the Algebra Survival Guide, and the 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.
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, as long as we are ganging up on Dear Aunt Sally, there's another critical issue we must address. The very thing that we are saying about multiplication and division also applies to addition and subtraction. This is so even though the "A" of "Aunt" (which stands for "Addition") precedes the "S" of "Sally (which stands for "Subtraction.").
This means that if a problem contains both addition and subtraction symbols, you don't necessarily perform the addition before performing the subtraction. Nor vice-versa. Rather, for addition and subtraction, you do whichever operation appears first as you read the expression from left to right.
EXAMPLE: In the problem: 10 – 2 + 6, you subtract 2 before you add 6, as follows:
10 – 2 + 6 = (10 – 2) + 6 = 8 + 6 = 14
Notice that if you incorrectly perform the addition first, you would get a different answer, as follows:
10 – 2 + 6 = 10 – (2 + 6) = 10 – 8 = 2 (incorrect answer)
How to make sense of all of this? Here's how I think of it. There are hierarchies in the math world just as there are hierarchies in the "real world." In the math world, one hierarchy deals with operations, and it states: multiplication and division are on the same level; they have the same power. Since these two operations are on the same level, there's no obvious way to tell which of those operation we should do first. So we will create a way. From here on out (this was said a long time ago!), if we want people to multiply first, we'll set up the expression in such a way that the multiplication symbol stands to the left of the division symbol. And vice-versa: if we want people to divide before multiplying, we'll set it up with the division symbol to the left of the multiplication symbol.
Now, since multiplication and division are on the same level, we can view multiplication and division as forming one hierarchy level, the multiplication/division level.
Similarly, another aspect of the math hierarchy states that addition and subtraction are on the same level. So those two operations form the addition/subtraction level.
However, there is a difference in hierarchy between the addition/subtraction level and the multiplication/division level. The multiplication/division level is higher, more powerful (in a sense), than the addition/subtraction level. That is why, given a problem containing, say, both multiplication and addition, you would always multiply before you add, no matter which term is to the left or to the right. But given a problem with multiplication and division, you need to use the left-right relationship to decide which of these "same-power" operations you do first.
One last question/answer unit. Suppose a problem has all four operations in it. How do you handle all of those operations? Here's what you need to know.
If a problem has all four operations, you perform the multiplication/division operations first. Then, once those have been performed, you perform the addition/subtraction operations. The following example illustrates this idea:
PROBLEM: 24 – 12 ÷ 2 + 3 x 3
24 – 12 ÷ 2 + 3 x 3 [problem]
= 24 – (12 ÷ 2) + 3 x 3 [set off division]
= 24 – 6 + 3 x 3 [do division]
= 24 – 6 + (3 x 3) [set off multiplication]
= 24 – 6 + 9 [do multiplication]
= (24 – 6) + 9 [set off subtraction]
= 18 + 9 [do subtraction]
= 27 [do addition]
Moral of the story: don’t let 'Dear Aunt Sally' fool you into thinking that you must do multiplication before division, or that you must do addition before subtraction. For each hierarchy level — multiplication/division or addition/subtraction — you do whichever operation comes first as you read the expression from left to right.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two teenage children. Josh is the author of the Algebra Survival Guide, and the 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.
At his blog, Josh writes about math education, offering tips and tricks. Josh also authors Turtle Talk, a free monthly newsletter with an engaging "Problem of the Month." You can see a sample issue here or subscribe at this site. 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 two fun math Qs/day.
