Arithmetic Magic - The Introduction
So you think you're not good with math. Hmmm.
Well that's something I hear all the time so you
shouldn't feel too bad. The truth is, many, many
people, at one time or another, have felt the same way,
and have experienced something which has made them
feel bad about math and numbers in general. So the
common complaint, "I was never good at math," or "I
hate numbers and math," is as common as the common
cold and sometimes just as annoying. Interestingly,
what I have discovered from working with students
through the years is that this feeling of inadequacy
often stems from some negative experiences with the
subject. This might very well be the case with you.
Perhaps you had a hard time in one of your math
classes and felt very frustrated by it. Perhaps you
struggled year after year because you could'nt
understand the material being taught, or because the
pace was too fast. Perhaps you didn't understand
because the subject matter was presented in a way that
did not get across to you. Sound familiar?
My concern with the above scenario is that,
uncorrected, this problem usually leads to feelings of
frustration and inadequacy toward math in general. A
likely—yet very undesirable—end result would be the
drawing of incorrect conclusions about yourself and
your abilities. Do these sound familiar?: "I'm just not
good at math and never will be," or "I'm just not
smart enough to do math," or more general and even
more crippling, "I'm just not that smart."
I cannot count how many times I have
heard such statements, the like of which only
serve to drain the energy and
stunt the educational gains of even the best of us.
Unfortunately, these statements crystallize into
attitudes and beliefs which become difficult to shake
off. However mistaken these beliefs might be, their
burdensome influence remains for a very long time—
sometimes even for life. The end result—math
illiteracy.
This is very unfortunate—even tragic—since the
ability to work with numbers and do math are skills
that have countless benefits. After all, no one should
be doomed to mathematical illiteracy, especially when
the alternative is so near within reach. Let's consider
the importance of basic math skills in everyday life.
For one, you couldn't count without math, pay bills
and then balance a checkbook, understand basic
financial matters, such as by how much money will
grow at different interest rates, or even figure how
much change you should get after a simple purchase
transaction. What a sorry state we would all be in if
we couldn't do these simple things! Now think for a
second. Suppose you could add, subtract, multiply and
divide numbers so that routine calculations were no
longer a bother. Numbers are now no longer your
enemies but your friends. A very likely consequence
would then be that the word math no longer triggers a
bunch of negative impressions but rather positive ones.
You now consider yourself good at a very tough
subject and therefore consider yourself a "cut above
the rest." The implications of this new self-appraisal
are enormous.
Furthermore, imagine having a simple method
which allowed you to figure percents easily. Then you
could figure discounts in a store and know exactly how
much less you would pay for items that go on sale. As
you will soon see, working with calculations involving
percents is nothing more than understanding decimals
and how to multiply numbers. Of course, rather than
figure the discounts yourself you could wait until you
got to the cash register to pay for the items. This
would always be a possibility, but once again these
methods do not only have one practical use. If they
did, their value would be that much less. Percents
figure in many other areas of the real world, not just in
purchase transactions. In short, these newly acquired
skills, whether they be in the area of multiplication and
addition, or in the area of percent problems, will
certainly open up all sorts of possibilities within the
mathematical realm and unleash capabilities within
you that you never thought you possessed. In short,
this is the aim of Arithmetic Magic: to foster an
appreciation of mathematics through the study of the
basic arithmetical operations.
With this goal in mind, many positive things are
achieved. Specifically, as you learn the techniques in
this book, your new ability to work with numbers will
definitely give you an advantage—an edge, sharp and
cutting—to whatever you do. Aside from giving you a
command over numbers, which will come in handy in
daily life and serve you countless times in your days
going forward, the techniques will pave the road
toward new-found confidence and a boost in attitude
towards your mental abilities. This—more than the
computational ability—is the key payoff. Personally,
it would be difficult for me to estimate the number of
times I have used these techniques or the number of
ways in which they serve me. In fact, I am constantly
coming up with new combinations of the techniques
and teaching both myself and my students new
methods based on existing ones. You will find
yourself doing the same as you read this book with an
open and hungry mind.
For those of you who think the ability to work
easily with numbers is beyond them, think again.
From my experience, I am convinced that the ability to
work well with numbers is not solely a byproduct of
inborn talent. This is in spite of what many might
think. Barring any severe mental abnormalities, an
individual can improve his ability at math. Even the
so-called "classified" individuals, whether they be
considered "learning disabled" or afflicted with ADD
(Attention Deficit Disorder)—whatever such
classifications might mean—have benefitted from
these techniques. Part of the inability to do math
seems to be simply that, when it comes to math and
working with numbers, many individuals concede
defeat before even trying. Because such people "throw
in the towel" before the fight is over and never attempt
to do the math, they reach the mistaken conclusion that
math and numbers are just not for them. An erroneous,
yet very sorry, conclusion.
Consequently, if you've been frustrated by math—
and its fiendish counterparts, numbers—it should be
refreshing to hear what I have discovered regarding
this particular illness: you can do math and you can
work with numbers. This is something I am sure of
and, by the time you've read this book, something I
will have proven to you. I can state this positively
from experience. What you need, and what is
absolutely essential, is the right approach. Time and
time again, I have found that many people cannot work
with numbers because they have simply convinced
themselves that they are bad at math. Why? It seems
that these unfortunate individuals experienced some
frustration with the subject early on. Because these
early failures have far outweighed any successes, these
same individuals come under the mistaken notion that
they cannot do math. This situation now sets in
motion a negative cycle. These early failures—or
"lack of successes"—snowball to the point of no return
and form a downward spiral. Once this happens, the
individual is then certain that math is not a fun subject
and one to be avoided at all costs. After all, you can't
do something if you believe you can't.
Admittedly, mathematics is not an easy subject.
Math has its roots in numbers and although the more
abstract branches of this subject might find themselves
far removed from numbers, the foundation in numbers
is still there. You cannot escape them. There is even a
whole branch of mathematics devoted to the study of
numbers and number properties called Number
Theory. It would seem then that the royal road to
understanding math lies in understanding numbers and
that the two were inseparable. I say royal because
even such famous mathematicians as Pythagoras—you
know the one with that famous theorem in geometry—
have written that numbers and counting formed the
foundation of all mathematics. Moreover, since
math—with its many branches and applications—
serves as one of the principle tools to explaining the
world around us, and since these things we call
numbers find themselves intricately woven into all of
mathematics, a good understanding of them can
certainly help pave the way to a better understanding
of mathematics in general. Even doing calculus
requires that you understand the basic arithmetic
properties of numbers. As though this were not all,
one very real and definite benefit in learning and
studying mathematics comes from the educational gain
made in rational and logical thinking. Now that's a
nice payoff!
What I have laid out here in four simple chapters is
the basis for understanding numbers. This is done
through a study of the four basic arithmetic operations:
addition, subtraction, multiplication, and division. As
you will soon see, each of these operations finds itself
related by special properties to the other. This
relationship allows us to simplify even further the
study of these basic operations. I do use some basic
math terminology but I deliberately avoid getting
weighted down in special names. Far more important
than memorizing any terminology is understanding and
learning the techniques and knowing how and when to
utilize the methods.
As someone who struggled with math—who understands
the frustrations and feelings of inadequacy
that this subject can rain upon you—I put this book
together with the confident hope that it will help you
gain a whole new perspective on math and numbers.
This new attitude might foster a desire for further
study, and this would ultimately lead to a higher rate of
math proficiency. I am delighted to tell you that in
overcoming my frustrations toward math and numbers,
I have benefited enormously from gains made in selfconfidence
towards this subject and many others. I
now approach any subject matter or challenge—
whether personal or academic—with a feeling of "I
can do it." If after reading this, your attitude towards
math and numbers has changed to a more positive and
less fearful one, then my goal will have been
accomplished.
One final note that should set you in a very positive
frame of mind before reading is the following: the
arithmetic techniques presented here are not only
empowering but also mind stimulating. They are
confidence boosting. Now I am quite aware that in the
high technology world of today, calculators are readily
available and found everywhere; even cellular
phones have these built-in devices ready to serve you.
But the truth of the matter is that even though
technology is everywhere, the use of such technology
is sometimes limited. The point of this
discussion is that there is something special about
knowing how to do something, and this is particularly
true when you can do something that others think is
hard. Such as mathematics. The other point is that
knowledge in and of itself has a very rewarding aspect
to it, one that often cannot be thoroughly evaluated or
appraised. So you are not learning these techniques
solely to do away with your calculator, as this device
will always have its uses. You are learning them so
that you come to understand that what you thought was
undo-able is actually do-able and what you thought
might be impossible is actually possible. This fact is
very empowering indeed and can serve as the very
bridge to higher achievement.
So let's get going and begin our journey into
Arithmetic Magic. May you never be the same again.
Teach Your Kids Arithmetic - Calculations with Percents
How important on a scale of 1 to 10 would you say knowing how to do basic percents was? Personally, I cannot think of a more important basic math function than this. Yet most people struggle with the very basics of percent calculations. This should not be the case. My aim in writing Arithmetic Magic was to show people that they can do basic arithmetic, and this once realized would empower them to study math at a higher level and become more interested in this most precious discipline.
Let's look at what a percent really means. Percent from the Latin literally means "per hundred." A percent is one part out of a hundred. Ten percent literally means "10 parts out of the hundred," or "10 parts per hundred." When we take a percent of a number, we are actually taking a portion of that whole. In other words, when we take 10% of a quantity, we are trying to calculate a portion of that quantity equal to 10 parts out of one hundred. If you read my article "Fractions, Percents, and Decimals," then you know that these three mathematical entities are one and the same thing: that is a percent is a decimal is a fraction. Consequently 10% is nothing more than 1/10 of the whole.
Now as a decimal, 10% is 0.1. To take 10% of any number we need only shift the decimal point of that number one place to the left. So 10% of 50 is equal to 5. (In a whole number the decimal point is not written but can be found immediately to the right of the last digit. Thus in 50, the invisible decimal point is after the 0, that is 50 can be written as 50. ) Based on this principle, and this principle alone, we have a way of calculating percents very easily. How you say? Let's get into this.
We use the following facts: 10% is 1/10 of a number; 1% is 1/100 of a number (a two decimal shift to the left) or 1/10 of 10%; and 5% is half of 10%. To get a percent of a number we use simple combinations of these facts. Watch.
Suppose you want to calculate 20% of 40. Now 20% is twice 10%. So get 10% and multiply by two: thus 10% of 40 is 4 and 20% is 8. Abracadabra. You want 21% of 40. No sweat. 20% is 8 (just done) and 1% is 1/10 of 10% of 40 which is 0.4 (a two decimal shift of the decimal in 40 to the left) Therefore 21% is 8.4. You need 35% of 50. Okay, 10% is 5 and 30% is three times this or 15; 5% is half of 10% or half of 5 or 2.5; put the pieces together and you have 35% of 50 is 15 + 2.5 or 17.5.
To show the sheer beauty of this method, let's do one more: calculate 28% of 62. We can do this several ways. Let's break down 28% into 2 x 10% + 8 x 1%. Now 10% of 62 is 6.2, and 20% is double this or 12.4; 1% is 0.62 and 8% is 8 times this or 4.96 (granted this is harder, and you probably need to study my techniques a bit more in-depth to do this. See "Arithmetic Magic" on my website); thus 28% of 62 is 12.4 + 4.96 or 17.36. Nice job!
So you see, working with percents is not difficult at all once you know the tricks. And you should know them since everyone works with percents. Now you can impress your boss with your new-found math skills.
Math Geeks are Cool
Why is it that math geeks are considered squares? That is, why is the straight-A math student considered Mr. Un-popular in school? It's funny how twisted things are in life. The cool kids in school are the ones smoking pot and cigarettes, perhaps taking other illegal substances, having sex with other students, and getting into at least one fight per quarter. Meanwhile, the straight-A kids are considered geeks, even though they are the ones that will build the country, write the future laws, and find the cures for the most vicious diseases that affect us all today.
With the world in such a chaotic state today, with morals decaying faster than you can say "moral turpitude," and with parents writhing at the thought of coping with their soon-to-be teenage son or daughter, it is no wonder that people of this generation are asking why things are so different from only two decades ago. Well the answer is not too hard to find once we know where to look.
One thing we need to do is examine the priorities in our culture. If you read my article "20/20 Speaks out about Education in America" then you can understand why a small seemingly insignificant country like Belgium could be excelling in education while America is lagging behind. In Belgium, math geeks are extolled—not vilified as some depraved life form. Our youth has distorted values and they worship figures in our culture who really should be given a lesser place within the hierarchy of worship. Now really should gangster rap stars be lauded as saviors of society? Hardly.
Unfortunately, a steady diet of MTV and the internet teaches children from an early age on, to worship such individuals and to give them role model status. (Hey parents, have you noticed the changes in your children's language over the years? Does it contain a lot of cursing? Well, try listening to a gangster rap song and then you'll know why. And rappers, don't give me that stuff that you're just talking about your life and reality. Come now. Don't you see what you're doing to the kids?)
Now take some fame, a couple hundred million dollars, clothing lines and other businesses to boot, and you have a ghetto kid thinking because he made it big in rap that he is a yardstick to be measured by. (And hey this observation does not go out to rap music only; such applies to other genres of music that are poisoning our kids' minds.) Do I believe that freedom of speech should be curtailed. No. Absolutely not. But I think we need to realign how we think about such moral decay in our youth, for they are our future.
Let's teach our children from early on that the math geeks and such should be lauded, for on their shoulders our very future rests. Let us adjust what our perceptions are on "cool," and "popular" and let us teach our children that straight-A's are cool and that pulling poor grades, smoking pot and cigarettes, and taking drugs are uncool. This could be the difference between a bright future and a dismal one. Heck, I'm already disappointed with the world I live in. Let's not make it any worse. Speak up on this issue. Don't be the silent minority, be the loud majority. God bless.
