## Where math comes alive

### 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.

### Reader Input on Slope Post

A longtime reader of Turtle Talk, Jeff LeMieux, of Oak Harbor, WA, sent in a suggestion based on today’s post on positive and negative slope. Jeff found a way to help students remember not only positive and negative slope, but also the infinite slope of vertical lines, and the 0 slope of horizontal lines … all using the letter “N.”

This is clearly a situation where the picture speaks more loudly than words, so I’ll just let Jeff’s submitted picture do the talking. By the way, to see this image even better, just double click it!

Slope Memory Trick

Thanks for putting this together and sharing it, Jeff!

### Remember the Difference in LOOK between Positive and Negative Slope

Some ideas just slap you in the face.

I got slapped this morning as I was flying home from LA to Albuquerque. Those little cocktail napkins they hand out with “beverage service” often give me the urge to write. So this morning, nerdily enough, as I sipped my orange juice at 30,000 feet above the Salton Sea, I worked on figuring out a better way to help students grasp the difference in look between positive and negative slope.

That’s when I got “slapped.”

First, you must realize that I use the three-letter abbreviations of POS and NEG for positive and negative. Do some of you use these as well? I mention this because those abbreviations hold the key. You have to use the first letter of the NEG abbreviation and the last letter of the POS abbreviation.

The first letter of NEG is, of course, “N.” But look what I noticed …

Visual Clue for Negative Slope

The trick for POS is a tad more complicated. But I’m hopeful it will work.

Visual Clue for Positive Slope

So what do you think? Will this work for your students?

If you test it out, please let me know what you find. I’m interested to know. Thanks!

### SAT and ACT Preparation

Every once in a while I get a call like this:

“Hi, this is ____. My son is taking the SAT on Saturday. Can you, like, help him get ready for the test so he can score real high?”

“Real high.” That’s what I wonder if the parent is.

I’ve been tutoring students to prepare them for the SAT and the ACT for many years now, and there’s one thing I’ve found to be true:  there is no shortcut to doing well.

While there are steps students can take to do better on these tests — including a few time-tested test-taking strategies — there is nothing I can do as a tutor (or a test-prep coach) that substitutes for solid academics over a student’s entire school career, and a longterm approach to the high-stakes tests.

What I’d like to do in this post, then, is to sketch out a road map that will help parents take a longterm approach to the SAT or the ACT.

3rd)  Have a lot of discussions with your children. Talk about intellectual issues, politics, science, math, literature, the news, etc.

4th)  Starting in 8th grade or around that time, find a list of 1,000 or so words that are on the SAT and the ACT and help your child learn 5 words a week. Use the words during discussions. Encourage your child to use his newly learned vocabulary. Help your child learn the nuances of the words. The best way to learn vocabulary is through hearing the words spoken correctly, and trying to use them yourself.

5th)  Starting in late 9th grade or early 10th grade, get your child a book on the ACT or on the SAT. Have your child start to take short practice tests. Help your child get familiarized with the tests.

6th)  Talk with your kids about the importance of these tests. Discuss how much good scores on the SAT or the ACT can open up doors to the best colleges and universities. Discuss the importance of scholarships and correlate high test scores with scholarship opportunities.

7th)  Toward the end of 10th grade, start to look for a good test-prep program. Or look for a tutor who can help your child start getting ready for these tests.

8th)  After your child has had experience taking some practice tests, sit down with your child and together set some goals for test scores. Make clear that the goals are being set just to give your child something to aim toward.

If you take these steps, your child will have a big head start toward doing well on the SAT or on the ACT. And you will never find yourself in the position of thinking that a couple of sessions before the test will “do the trick.”

### Why the LCM trick works

This video explains why the trick for finding the LCM works. Some people asked to explain the math behind the trick, so here it is.

### Divisibility by 2, 5 and 10

Here are the tricks for divisibility by 2, 5 and 10.

### Find the LCM — FAST!

Here’s a video that goes with a blog entry that many people have found helpful: Find the LCM — FAST! This trick can be a real time-saver, so feel free to pass this around.