Kiss those Math Headaches GOODBYE!

Posts tagged ‘Dear Aunt Sally’

Please Revise My Dear Aunt Sally


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.

 

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Addition and Subtraction: More Bad Behavior by Dear Aunt Sally


Attention:  Dear Aunt Sally may not be fit for teaching students algebra!

A problem has been discovered in sweet Aunt Sally’s little memory trick:  Please Excuse My Dear Aunt Sally.

Actually, make that two problems.

The first, revealed in my 9/9 post below, is that Aunt Sally wrongly makes students think they’re supposed to multiply before dividing. That’s because the word My (standing for MULTIPLY) comes before Dear (standing for DIVIDE).

Countless students have been deceived into thinking they’re supposed to multiply before dividing [See the 9/9 post for the full run-down on this problem.]

Today I want to point out another problem, and offer two solutions.

The second problem is that, since “Aunt” (standing for ADD) comes before “Sally” (standing for SUBTRACT), countless other students have been led to think they are supposed to ADD before they SUBTRACT.

Well, what are students supposed to do?

First of all, students need to realize that adding and subtracting are at exactly the same level of hierarchy as each other. But if that’s true, how can students ever decide which to do first.

Easy! Same solution as with multiplying and dividing. We simply look to see which of these  operations is written first as we read the problem left to right.

Example:  in the expression  8 + 3 – 4, the addition symbol precedes the subtraction symbol, so here we add before subtracting. And we simplify the expression like this:

8 + 3 – 4
=  11 – 4
= 7

But in the expression   8 – 3 + 4
the subtraction symbol is written before the addition symbol, so here we subtract before we add, and we simplify the expression like this:

8 – 3 + 4
=  5 + 4
=  9

It’s really that simple. Pay attention to which operation sign comes first as you read the problem from left to right. Then do the operations in the correct order based on that.

One other solution:  in my book, the Algebra Survival Guide, I get away from the Please Excuse My Dear Aunt Sally approach, as I create my own memory trick, one that involves Strawberry Mousse. If you want to take a look at this approach, check out my book at this site.

From the homepage, click the link that says:  View Sample Chapters of the Algebra Survival Guide, and download the chapter on Positive and Negative
Numbers.

Cover of

Cover via Amazon

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Is Dear Aunt Sally “Batty”?


When tutoring, I enjoy pondering the mistakes students make. I find mistakes interesting to think about, as they give me insights into why students have trouble with math in general.

And one of the mistakes I’ve been seeing early this year involves one of our most colorful characters from the world of algebra, Dear Aunt Sally. As in:  “Please Excuse My Dear Aunt Sally,” the mnemonic phrase designed to instill an understanding of the order of operations.

I’ve never found out what Dear Aunt Sally did that requires us to excuse her poor behavior. But I have discovered something that might qualify for bad behavior. It’s the way in which the words of this very expression sow confusion for many students.

In particular, I’m referring to the fact that the “M” of “my” appears to come before the “D” of “dear.” And the fact that therefore, many students conclude that they must always do multiplication before division.

Generally, when being tutored, students take me at my word. I mean, I do have a good reputation, and I’ve written a few math books, so for the most parts, kids give me the benefit of the doubt, if I’m telling them something they have not heard before (it happens).

But when it comes to “dear Aunt Sally,” and the fact that I sometimes need to clear up their confusion about her, boy do kids get defensive, as if Aunt Sally was really their aunt, and they need to make sure I don’t hurt her feelings …?

I get looks like, “What do you mean I’m doing it wrong?” And “Are you sure, Josh?” And “Are you really sure, Josh? because my teacher … ”

Since Aunt Sally is such a “dear,” people tend to take her at face value. But too much.

So here, let me, the “math ogre” in this respect, set the record straight.

Just because the “M” of “my” seems to come before the “D” of “dear”, that does NOT mean that we do multiplication before division.

The rule actually is this:  you do not necessarily do multiplication before division; and you do not necessarily do division before multiplication.

What you do is this: if an expression has both multiplication and division in it, you do WHICHEVER OF THOSE TWO OPERATIONS COMES FIRST AS YOU READ THE EXPRESSION FROM LEFT TO RIGHT.

So, if you have this expression:  12 x 4 ÷ 6, you WOULD work out the multiplication before the division, but ONLY BECAUSE the multiplication symbol comes before the division symbol as you read the expression from left to right. So this expression should be simplified like this:

12 x 4 ÷ 6
=  48 ÷ 6
=  8

On the other hand,  if you have this expression:    12 ÷ 4 x 6, you WOULD NOT do the multiplication first. Rather, you would do the division first because the division symbol comes BEFORE the multiplication symbol as you read the expression from left to right.

So this expression would be simplified like this:

12 ÷ 4 x 6
=  3 x 6
=  18

IMPORTANT:  Notice that the way you work out the expression can make a difference. For example, if you simplified the last expression incorrectly, you would get a different answer. This is wrong, but I am going to do the multiplication before division, like this:

12 ÷ 4 x 6
=  12 ÷ 24
=  1/2

So bear in mind that you can and will get the wrong answer if you don’t follow the true rule.

Moral of the story:  don’t let Dear Aunt Sally fool you into thinking that you must do multiplication before division. You do whichever operation comes first as you read the expression from left to right.  And you continue doing operations in the order that they appear from left to right.

One last point: you might be wondering why mathematicians have made the rule the way it is rather than the way students get fooled into thinking it works.

The reason, I believe, is so we have flexibility when we write expressions. If we want someone to do division first, we write the division part of the expression first;  if we need someone to do multiplication first, we write that part of the expression first.

If the rule really stated that you always do multiplication before division, there would be no way to write an expression with both operations in such a way that the division  is done first. That would hamstring people in writing math expressions, and we mathematicians cannot tolerate being limited in that way.