Kiss those Math Headaches GOODBYE!

Posts tagged ‘Multiplication’

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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Multiplication Trick — Multiplying by 5


Time for a math trick …

Q:  How do you multiply an even number by 5 in lightning speed?

A:  Divide the number by 2, then tack on a “0.”

Example:   5 x 24

Divide 24 by 2 to get 12.

Tack a “0” onto 12 to get 120. Presto, nothing up your sleeve. It’s that easy.

Why does it work? Hint: Think about how we multiply by 10. Then think about how multiplying by 5 compares to multiplying by 10.

Rotated version of File:Symbol support2 vote.svg.

Image via Wikipedia


Try these for fun (answers at bottom of post):

a)  5  x  16

b)  5  x  8

c)  5  x  28

d)  5  x  64

e)  5  x  142

f)  5  x  2,468

g)  5  x  6,042

h)  5  x  86,432

j)  5  x  888,888


Answers:

a)  5  x  16 = 80

b)  5  x  8 = 40

c)  5  x  28 = 140

d)  5  x  64 = 320

e)  5  x  142 = 710

f)  5  x  2,468 = 12,340

g)  5  x  6,042 = 30,210

h)  5  x  86,432 = 432,160

j)  5 x 888,888 = 4,444,440

k)  5  x  2,486,248 = 12,431,240

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.

Multiplication Trick #5 — How to Multiply Two-Digit Numbers by 11


This is the fifth in my series on multiplication tricks. I suggest that you make mental math “tricks” a steady part of your math instruction. Benefits students will reap include:

—  delight with the tricks themselves

—  enhanced confidence in working with numbers

—  students who otherwise don’t like math — or don’t like it much — often find the tricks irresistibly fun and interesting

TRICK #5:

WHAT THE TRICK LETS YOU DO: Multiply two-digit numbers by 11.

HOW YOU DO IT:  To multiply a two-digit number by 11, first realize that the answer will have three digits. The first (left-most) digit of the answer is the first digit of the number; the last (right-most) digit of the answer is the last digit of the number; and the middle digit is the sum of the first and last digits.

But those are just words … here’s a living, breathing example …

Example:  11 x 25

 

Look at 25. The first digit is 2; the last digit is 5.

First digit of answer is 2, so thus far we know the answer looks like:  2 _ _

Last digit of answer is 5, so now we know the answer looks like:  2 _ 5

Middle digit is 7, since 2 + 5 = 7.

The answer is the three-digit number:  2 7 5, more casually known as 275.

It’s that easy!

ANOTHER EXAMPLE:  11 x  63

First digit of answer is 6, so thus far we know the answer looks like:  6 _ _

Last digit of answer is 3, so now we know the answer looks like:  6 _ 3

Middle digit is 9, since 6 + 3 = 9.

The answer is the three-digit number: 6 9 3, or just 693.

Try these for practice:

11 x 24

11 x 31

11 x 52

11 x 27

11 x 34

11 x 26

11 x 62

 Answers:

11 x 24 = 264

11 x 31 = 341

11 x 52 = 572

11 x 27 = 297

11 x 34 = 374

11 x 26 = 286

11 x 62 = 682

NOTE:  If you’re clever (and we’re sure that you are), you have probably realized that this trick, as described, works only when the digits add up to 9 or less. So what do you do when the digits add up to 10 or more? Some of you may figure this out on your own. For those who need a little help, the answer to this will be included in an upcoming blog post.

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly  like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com  Just click the links in the sidebar for more information! 

Multiplication Trick #2 — How to Multiply by 15 FAST!


Here’s the second in my set of multiplication tricks. (The first was a trick or multiplying by 5.)

TRICK #2:

WHAT THE TRICK LETS YOU DO: Multiply numbers by 15 — FAST!

HOW YOU DO IT: When multiplying a number by 15, simply multiply the number by 10, then add half.

EXAMPLE:15 x 6

6 x 10 = 60

Half of 60 is 30.

60 + 30 = 90

That’s the answer:15 x 6 = 90.

ANOTHER EXAMPLE:15 x 24

24 x 10 = 240

Half of 240 is 120.

240 + 120 = 360

That’s the answer:15 x 24 = 360.

EXAMPLE WITH AN ODD NUMBER:15 x 9

9 x 10 = 90

Half of 90 is 45.

90 + 45 = 135

That’s the answer:15 x 9 = 135.

EXAMPLE WITH A LARGER ODD NUMBER:23 x 15

23 x 10 = 230

Half of 230 is 115.

230 + 115= 345

That’s the answer:15 x 23 = 345.

PRACTICE Set:(Answers below)

15 x 4

15 x 5

15 x 8

15 x 12

15 x 17

15 x 20

15 x 28

ANSWERS Set:

15 x 4=60

15 x 5=75

15 x 8=120

15 x 12=180

15 x 17=255

15 x 20=300

15 x 28=420

 

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly  like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com  Just click the links in the sidebar for more information! 
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Memorizing those Times Tables


Eegads! The times tables.

Is there any area of elementary math more fraught with stress and anxiety, save, perhaps, long division? Probably not. But for good reason.

Despite what a tiny minority of conceptual-learning purists might say, the times table facts ARE critical. Let’s face it: you really DON’T want your children to spend the rest of their lives reaching for the calculator to figure out 6 x 7; a certain amount of math simply needs to become automatic, to allow students to succeed at higher math skills and and to gain higher math concepts. Not only that, but knowing the times tables is widely recognized as a crucial milestone in children’s elementary math development.

In my work as a tutor, I’ve used many approaches to teach the times tables over the years, and each of them has one benefit or another. But I’ve settled on one technique as my “old-faithful” approach. This technique combines elements of both play and discipline, and it also melds both the “conceptual” approach and the “pure memorization” approach.

This technique relies on a three-step process, and it’s easy to learn and teach.

The first step is to simply isolate a particular times table fact set you’d like your child to work on, for example, the 4s. This act of isolating itself is critical. The child knows that she or he is required to memorize a limited set of facts for now (not the entire times tables), and that narrowing of the task decreases anxiety.

Once you’ve settled on the fact set, the second step begins, and it can be quite fun. In this second step there should be no mention even made of the times tables. All you’re doing in this step is laying the foundation for times tables facts. What you do here is work with your students/children to help them learn to first COUNT UP by the number you’re dealing with. So for example, if you’re teaching the 4s, you simply teach children how to COUNT UP by 4s. What that means is that you teach your children how to think their way through knowing and saying the following with speed and ease:  0 – 4 – 8 – 12 – 16 – 20 – 24 – 28 – 32 – 36 – 40 – 44 – 48. 

I’ve found that most children take well to this learning process if you approach it in the spirit of a game. You might, for example, start by saying 0 and then throw your child a ball. She or he will then say 4 and throw the ball back to you. You then would say 8, and then throw the ball back to your child. Keep going till you hit the peak number, 40, 48, or wherever you decide to stop. 

Another way to make this into a game for young children is to make it into a game like “patty-cake.” Make up a set of hand gestures to which you, very quietly, say:  1-2-3, and then clap hands and loudly say “4!” Then use the same hand gestures to quietly say:  5-6-7, and then clap again and loudly say: “8!” There are many ways to make this process of counting by 4s game-like. And if you’re short on ideas, ask your children/students what would make it fun for them.

In any case, once your children can accurately COUNT UP by 4s, work with them in the same fashion to COUNT DOWN by 4s. Same idea, but now you start by saying 48, or 40, and then help them count DOWN:  44 – 40 – 36 – 32 –  28 – 24 – 20 – 16 – 12 – 8 – 4 – 0. This takes a bit more time, but it can be done — and more easily than you might imagine.

Once your child can count both up and down, she or he has the mental “scaffolding” on which the times table facts are hung, as it were.

And so the third step involves combining this “scaffolding” with the actual times tables. Here’s how.

Have your children memorize what I call THE THREE KEY MULTIPLICATION FACTS:
 x 1,  x 5, and x 10.

For example, when learning the 4s, these key facts would be:
4 x 1 = 4
4 x 5 = 20
4 x 10 = 40

Once children memorize those three key facts, help them see that to find 4 x 2 and 4 x 3, they just COUNT UP by 4 once or twice, beyond the key fact of 4 x 1 = 4. Similarly, to find 4 x 6 and 4 x 7 they just COUNT UP by 4 once or twice, beyond the key fact of 4 x 5 = 20. And to find 4 x 11 and 4 x 12, they just COUNT UP by 4 once or twice beyond the key fact of 4 x 10 = 40. 

Work on this first, and have them master it before proceeding.

Once a child knows these facts, she or he has 9 of the 13 key facts (going from 4 x 0 through 4 x 12).

To learn the four other facts, help children see that to find 4 x 4 and 4 x 3, they just COUNT DOWN by 4 once or twice, below the key fact of 4 x 5 = 20. And to find 4 x 9 and 4 x 8, they just COUNT DOWN by 4 once or twice, below the key fact of 4 x 10.

By breaking the process of learning the times tables into these steps, you make the process less daunting for children. By teaching students how to COUNT UP or COUNT DOWN by the number you’re learning, you help children develop many rich aspects of number sense. And by connecting the process of COUNTING UP or DOWN to the times tables, you help children learn these critical facts both solidly and with understanding.

My advice:  try it. I guarantee that you’ll like it.

Happy Teaching,

—  Josh


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Multiplication Trick #1 — Fun with the 5s


 

SPICE IT UP!

That’s my advice to teachers and parents who see students getting bored or frustrated as they try to learn their times tables. 

As you help students learn these critical facts, it helps, from time to time — to work on multiplication in a fun and relaxing way.

This is the first in a series of blogposts that make it more pleasurable to learn multiplication facts — by teaching multiplication tricks. Each post will contain a complete lesson plan:  instruction, practice problems, and all answers. 

The first such trick is for multiplying by 5.

TRICK #1:


WHAT THE TRICK LETS YOU DO:
  Multiply numbers by 5.


HOW YOU DO IT (EVEN NUMBERS):
When multiplying an even number by 5, just take half the value of the even number, then put 0 at the end.  Ta da … that’s your answer.

 

Example:  5 x 14

Half of 14 is 7.

Put down the 7, then put a 0 after it, and you get 70.

That’s the answer:  5 x 14 = 70.

Can you believe that it’s that easy? Watch how you can do the same feat with larger numbers…

 

Another example:  5 x 48

Half of 48 is 24.

Put down the 24, then put a 0 after it, and you get 240.

That’s the answer:  5 x 48 = 240.

 

PRACTICE Set A:  (Answers at bottom)

5 x 8

5 x 16

5 x 4

5 x 28

5 x 36

5 x 84

5 x 468


HOW YOU DO IT (ODD NUMBERS):
When multiplying an odd number by 5, first subtract 1 from the odd number, thus making it an even number. Then use the trick (above) for even numbers. And here’s the new thing to know — instead of putting a 0 after the result, put a 5.


Example:  5 x 13

13 – 1 = 12

Half of 12 is 6.

Put down the 6, then put a 5 after it, and you get 65, That’s the answer:

5 x 13 = 65.

 

Another example: 5 x 29


29 – 1 = 28

Half of 28 is 14.

Put down the 14, then put a 5 after it, and you get 145. That’s the answer:

5 x 29 = 145.

 

PRACTICE Set B:  (Answers at bottom)

5 x 7

5 x 13

5 x 9

5 x 15

5 x 23

5 x 47

5 x 685

 

ANSWERS Set A: 

5 x 8  = 40

5 x 16  = 80

5 x 4  =  20

5 x 28  =  140

5 x 36  =  180

5 x 84  =  420

5 x 468  =  2,340

 

 

ANSWERS Set B: 

5 x 7  =  35

5 x 13  =  65

5 x 9  =  45

5 x 15  =  75

5 x 23  =  115

5 x 47  =  235

5 x 685  =  3,425

 

 


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