## Kiss those Math Headaches GOODBYE!

### 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.

### How to tell if a Number is Divisible by 8

I’ve explained a number of divisibility rules lately, offering tricks to tell if numbers are divisible by 2, 3, 4, 5, 6 and 7.

There is also a trick for divisibility by 8, and that’s what I’d like to explain in this post.

Essentially the trick for 8 is a lot like the trick for 4. If you’d like to refresh your memory on how that trick works, just go here. (more…)

### How to tell if 4 Goes into a Number — Divisibility by 4

My last post offered a neat trick for seeing if 3 divides evenly into a number.

In this post, I’ll do the same thing for the number 4.

But my approach will be a bit different in this post. Instead of just presenting the “trick,” I will help us grasp the logic behind the trick by looking at two principles of divisibility. I’m doing this because learning the principles should boost your ability to work — or should I say, play? — with numbers.

First, a question: If a number divides evenly into one number, will it divide evenly into all multiples of that number? Example, given that 6 divides evenly into 30, will 6 divide evenly into the multiples of 30, such as 60, 90, 120, 150, etc. The answer is YES. This is a basic principle of divisibility, and we’ll call it the Divisibility Principle of Multiples, or just DPM, for short.

Second, related question:  if a number divides into two other numbers evenly, will it also divide evenly into the sum of those numbers? Check this out with an example, and see if it agree with your mathematical “common sense,” aka “number sense.”

4 goes into both 20 and 8, right? So does that mean that 4 goes into the sum of 20 and 8, namely 28? Well, yes, 4 does go into 28 evenly, seven times in fact.

Test one more example with  larger numbers. 9 goes into 90 and 36, right? So does that mean that 9 also must go into 90 + 36, which is 126? Yes again. This idea harmonizes with “number sense,” and it is in fact true. And we will use this soon. We’ll call this the Divisibility Principle of Sums, or just DPS.

To get started thinking about divisibility by 4, let’s consider one nice thing about 4:  it divides evenly into a number that ends in 0,  the number 20! This is helpful because in our base-10 number system, numbers that end in 0 are “friendly” — they fit into the system neatly.

Using DPM, then, since 4 goes into 20, it goes into all the multiples of 20:  20, 40, 60, 80, and  yes, 100! Why is this a big deal? Since 4 goes into 100, we can use DPM again to say that 4 goes into all multiples of 100:  200; 300; 400;  … 700; 1,300;  2,300, … we can even be certain that 4 goes into 6,235,700 since this is a multiple of 100 [100 x 62,357  =  6,235,700]

The implication of this is major:  if we want to figure out if 4 goes into any whole number, we can ignore all but the last two digits. In other words, to figure out if 4 goes into 5,296 we need only ask: does 4 go into 96. The reason is that we already know that 4 goes into 5,200, and using DPS, if 4 goes into both 5,200 and 96, we can be certain that 4 will go into 5,296.

So we now have the first part of our trick for 4:  To find out if 4 goes into any number, look only at the last two digits.

That’s a great start. But we can get even more precise.

First ask:  before 4 goes into 20, what other numbers does 4 divide into? Simple, 4 goes into 4, 8, 12, and 16.

DPS, we recall, says that  if a number, let’s call it n, goes into two other numbers — call them a and b — then n goes into their sum:   a + b.

We can use this idea right here. Since 4 divides into 20, and it also divides into 4, 8, 12 and 16, DPS guarantees that 4 also goes into the bold numbers below:
20 + 4 = 24
20 + 8 = 28
20 + 12 = 32
20 + 16 = 36

Big deal, you say, since you already knew this from the times tables.  True, but  going up one multiple of 20, you can start to see the power of this idea.

Since 4 divides into 40, and into 4, 8, 12 and 16, 4 also goes into the bold numbers:
40 + 4 = 44
40 + 8 = 48
40 + 12 = 52
40 + 16 = 56

Once again, since 4 divides into 60, and into 4, 8, 12 and 16, 4 also goes into:
60 + 4 = 64
60 + 8 = 68
60 + 12 = 72
60 + 16 = 76

Using the same pattern, we see that 4 goes into:  80, 84, 88, 92 and 96.

Great, you might say, this shows us a pattern, but not a “trick.”
Where is this long-promised trick?

What we need to realize is that the pattern leads to a trick.

For the trick, here’s what you do:

1st) Take the two digits at the end of any whole number.

2nd) Find the lesser but nearest multiple of 20, and subtract it from the two-digit number.

3rd) Look at the number you get by subtracting. If it’s a multiple of 4, then 4 DOES got into the original number. If it is NOT a multiple of 4, then 4 does NOT go into the original number.

Words, words, words, right? Let’s see some examples to give the words some life!

EXAMPLE 1:
Does 4 divide into 58?

PROCESS:
—  Nearest multiple of 20 to 58 is 40.
—  58 – 40 = 18
—  18 is NOT a multiple of 4, so 4 does NOT divide evenly into 58.

EXAMPLE 2:

Does 4 divide into 376?

PROCESS:
—  Focus on the last two digits:  76
—  Nearest multiple of 20 to 76 is 60.
—  76 – 60 = 16
—  16 IS a multiple of 4, so 4 DOES divide evenly into 376.

EXAMPLE 3:
Does 4 divide into 57,794?

PROCESS:
—  Focus on the last two digits:  94.
—  Nearest multiple of 20 to 94 is 80.
—  94 – 80 = 14
—  14 is NOT a multiple of 4, so 4 does NOT divide evenly into 57,794.

Make sense? If so, then you are ready to do some serious divisibility work with 4. Here are some practice problems, and their answers.

PROBLEMS:  Tell if 4 divides evenly into the following numbers.

a)   74
b)  92
c)   354
d)   768
e)  1,596
f)   3,390
g)  52,472
h)  831,062
i)  973,236
j)   17,531,958

a)   74:  74 – 60 = 14.  4  does NOT divide evenly into 74.
b)  92:  92 – 80 = 12.  4 DOES divide evenly into 92.
c)   354:  54 – 40 = 14.  4 does NOT divide evenly into 354.
d)   768:  68 – 60 = 8.  4 DOES divide evenly into 768.
e)  1,596:  96 – 80 = 16.  4 DOES divide evenly into 1,596.
f)   3,390:  90 – 80  = 10.  4 does NOT divide evenly into 3,390.
g)  52,472:  72 – 60 = 12.  4 DOES divide evenly into 52,472.
h)  831,062:  62 – 60 = 2.  4 does NOT divide evenly into 831, 062.
i)  973,236:  36 – 20 = 16.  4 DOES divide evenly into 973,236.
j)   17,531,958:  58 – 40  =  18.  4 does NOT divide evenly int0 7,531,958.

### 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.