## Kiss those Math Headaches GOODBYE!

### Secrets of the Calculator, Part 1

In my tutoring I am continually surprised by how little most students know about their calculators.

It is true that most students know the basics:  the four operations, the exponent key, the square root key, the Pi key, and maybe some trig fundamentals, like sin, cos 60 and tan. But aside from these basic keys and keystrokes, many students have little to no idea what the other keys do.

The funny thing is that there are so many keys and keystrokes that students would just “love” if they only knew about them.

So to help students out a bit here, I’m starting an occasional series whose name is just below.

### 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.

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

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

### Times Tables, Learning the Threes

What’s more important for early math than knowing the times tables?

Not much, right?

Since the times table facts are so fundamental, and because many students struggle with them, I’d like to share a strategy I came up with today for
learning the 3s. This technique works particularly well with students
who struggle with memorizing apparently random facts. (We know these
facts are not random, but if learned with nothing more than flash cards,
they can appear random.)

The strategy involves three stages, each stage bringing the child closer
to being able to QUICKLY access the desired multiplication facts. Here are the stages, in order they should be taught.

STAGE ONE:  “Patty-Cake Threes”

What I do here amounts to a “patty-cake” approach to learning the threes, which works like this.

The student and I sit facing each other with our hands up. We hit our right hands together and say “one,” then hit our left hands together and say “two.” Then we
hit BOTH HANDS TOGETHER and say, “THREE.” When saying the “one” and “two,” we utter the numbers quietly. But when we say “THREE” and all successive multiples of three, we say these numbers loudly, almost (but
not quite) shouting.

After three, we continue:  “four, five, SIX … seven, eight, NINE, ten,
eleven, TWELVE … ” and so on. So this gives children a fun way to
hear — and get a feel in their body for — the multiples of three, in the proper
order.

Image by davie_the_amazing via Flickr

STAGE TWO:  “Finger-Drumming”.

After the child has the rhythm of the number three, from the “patty-cake” approach, we do “finger-drumming.” To “finger-drum” the multiples of 3, the child makes a fist with one hand, and shakes it, saying with each shake, “one, two, THREE!” And when saying “THREE,” the child extends one finger from the fist. The child continues: “four, five, SIX,” and at “SIX,” he extends another finger, so he has two fingers out.

Then you ask the child, for example, “What is three times two?” Answer: the number he just said, “six.”

In this way, the child can “finger-drum” out all of the multiples of three. To
reinforce the times tables as you go, ask questions like:  “What is 3 x 4? What is
3 x 5? etc.” Each time you ask, the child must “finger-drum” till s/he gets the
correct answer. This flows very nicely from the “patty-cake” approach as it
builds on the rhythmic feel for counting in threes.

STAGE THREE:   “Finger-Skip-Counting”   The third stage follows “finger-
drumming.” To begin finger-skip-counting, the child must have done enough “finger-drumming” so s/he is quite familiar with the multiples in the correct order.

To “finger-skip-count,” 3 x 4, for example, the child holds out a fist and
runs through the multiples of 3, like this:  “Three (extending one finger), Six (extending two fingers), Nine (extending three), Twelve (extending four fingers).”  You ask, “So what is 3 x 4?” And the child answers:  “3 x 4 equals 12.”

I found it helpful to first just challenge the child with the multiples from 3 x 1 through 3 x 5. Once s/he develops competence there, proceed to “finger-skip-counting” the multiples from 3 x 6 through 3 x 10. Finally do 3 x 11 and 3 x 12.

Put all together, these three stages offer a fun and rhythmic way for children
to learn their multiples of three. I’m curious to find out if I can use a similar
approach for the 4s, and I’ll find out soon.

I can’t be sure, but it seems like children could probably learn their 4s
by jumping rope, or doing other activities with a rhythmic nature.

If any of you have used an approach like this one for learning the times
tables, feel free to share it.

### Abbreviating the Order of Operations

My recent posts about “Dear Aunt Sally” have, I hope, shown how dangerous it is to teach the memory trick of Please Excuse My Dear Aunt Sally — at least without some additional explanation.

Today I propose a way to save Dear Aunt Sally, for those of you who still like her.

As you know, the memory sentence is often abbreviated PEMDAS, which stands for:  PARENTHESES, the EXPONENTS, then MULTIPLICATION, then DIVISION, then ADDITION, then SUBTRACTION.

The problem with PEMDAS is that it makes kids think they always multiply before dividing, and that they always add before subtracting.

For students attached to PEMDAS, I let them use it, but I have them write it a novel way, so they realize they must pay attention to the left-right orientation of the operation symbols.

In the new way of writing PEMDAS, I put M and D in the same place, separate by the word “or.” Then I do the same for the A and S. So the whole memory device looks like this:

Alternative for PEMDAS

My suggestion is that teachers who like PEMDAS try this and see if your students start making fewer mistakes with the order of operations.

For those of you who never liked PEMDAS on the first place, I recommend that you check out the order of operations as presently in a clearer way, as it is in my Algebra Survival Guide. To get a feel for that, you can download the chapter of the book on Positive and Negative numbers here. Just click where it says:  See Sample Chapters of the Algebra Survival Guide and Workbook.

Then you’ll want to get the Survival Guide for the chapter on Order of Operations, which you can do through Amazon.com, here. Sorry, I can’t make everything available for free … I do have a business to run, with new products I’d like to create and make available.

Best way to write PEMDAS

### Multiplication Trick: x 25

This is a simple trick that anyone can easily learn. It is just a trick for
multiplying a number by 25.

If someone asked you what 25 times 36 equals, you’d probably be tempted
to reach for a calculator and start punching buttons. But remarkably, you’d
probably be able to work it out even faster in your head.

Since 25 is one-fourth of 100, multiplying by 25 is the same thing as
multiplying by 100 and dividing by 4. Or, even more simply:
first divide by 4,

Here’s the example:

Problem: 36 x 25
First divide 36 by 4 to get 9.
Then add two zeros to get: 900.
That, amazingly enough, is the answer.

Another example: 88 x 25
First divide 88 by 4 to get 22.
Then add two zeros to get: 2,200.

a) 25 x 12
b) 25 x 28
c) 25 x 48
d) 25 x 60
e) 25 x 84
f) 25 x 96

a) 300
b) 700
c) 1,200
d) 1,500
e) 2,100
f) 2,400

But, you say, what if the number you start with is not divisible by 4.
No problem. Just use this fact:
if the remainder is 1, that is the same as 1/4 or .25
if the remainder is 2, that is the same as 2/4 or .50
if the remainder is 3, that is the same as 3/4 or .75

So take a problem like this: 25 x 17
dividing 17 by 4, you get 4 remainder 1.
But that is the same as 4.25
Now just move the decimal right two places (same as multiplying by 100)

Another example: 25 x 18
dividing 18 by 4, you get 4 remainder 2.
But that is the same as 4.50
Now move the decimal right two places.

Another example: 25 x 19
dividing 19 by 4, you get 4 remainder 3.
But that is the same as 4.75
Now move the decimal two places to the right.

A) 25 x 21
B) 25 x 26
C) 25 x 35
D) 25 x 42
E) 25 x 63
F) 25 x 81

A) 525
B) 650
C) 875
D) 1,050
E) 1,575
F) 2,025

### 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

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!

### NOVEMBER PROBLEM OF THE MONTH

The problem:

It’s your friend’s 73rd birthday. You’ve put together a surprise party and baked a special coconut meringue cake. But at the last minute you realize that — golly gee! — you forgot to get candles.

Rummaging through your drawers with just five minutes before your friend’s scheduled arrival, you find that you do have 14 candles. And being a brilliant mathematician, you realize that you can represent the number 73 with these 14 candles, using every candle. How do you do it?

As a hint, here’s a model showing how to do a problem like this, if you are celebrating someone’s 44th birthday, when you have just 13 candles. Notice that each dot on the top row is one candle.

Note that you may use icing to create the symbols: +, –, x, ÷, and you may also put in exponents, using candles to show the value of the exponent.

Have fun!