Dr. Math Gets You Ready for Algebra: Learning Pre-Algebra Is Easy! Just Ask Dr. Math!

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Dr. Math Gets You Ready for Algebra

John Wiley & Sons

Chapter One

Operations are the arithmetic skills introduced and practiced in
elementary school. The fundamental operations are addition, subtraction,
multiplication, and division. Exponentiation is also an
operation. In algebra, the fundamental operations are as important
as they are in arithmetic. In fact, if you ever want to check your
algebraic work by substituting a number for the variable, you'll be
reminded of the arithmetic exercises that look more familiar.

Clive and Carissa have a lot of questions about what they're
learning. In this part, Dr. Math explains

Introduction to algebraic thinking

Variables

Exponents

Large and small numbers

Order of operations

Distributive property and other properties

Introduction to Algebraic Thinking

Algebraic thinking is the bridge between arithmetic and algebra.
Representing, analyzing, and generalizing a variety of patterns with
tables, graphs, words, and, when possible, symbolic rules are all
part of thinking algebraically.

What Is
Algebra?

Dear Dr. Math,

What is algebra

Yourstruly,
Clive

Hi, Clive,

Algebra is like arithmetic, but in algebra some of the numbers have
names instead of values. For example, if I ask you something like

3 + 4 · 5 - 6 ÷ 3 = ?

you can just apply the operations in the correct order to get

3 + 4 · 5 - 6 ÷ 3 = ?
3 + 20 - 6 ÷ 3 = ?
3 + 20 - 2 = ?
21 = ?

Now, suppose that instead I ask you something
like

(x + 3) · (x + 4) = 42

You can't apply the operations because you
don't know the values of the numbers. What's
x + 3? It depends on the value of x, doesn't it?

In this case, you might start guessing possible
values for x that would make the equation
true:

x = 1? (1 + 3) · (1 + 4) = 4 · 5 = 20 (No.)
x = 2? (2 + 3) · (2 + 4) = 5 · 6 = 30 (No.)
x = 3? (3 + 3) · (3 + 4) = 6 · 7 = 42 (Yes!)

However, suppose the problem changes to

(x + 3) · (x + 4) = 35.75

Now it becomes a lot harder to guess an answer. Algebra gives you
a set of tools for figuring out the answers to problems like this without
having to guess.

This becomes more and more important as you start using more
complicated equations involving more than one variable.
-Dr. Math, The Math Forum

What is
Algebraic
Thinking?

Dear Dr. Math,

How do you start to think algebraically?

Sincerely,
Carissa

Hi, Carissa,

Well, you already know about multiplication, division, addition, and
subtraction. One day (a long, long time ago) somebody-let's call
him or her Pat-who knew all of those things was sitting around
thinking about addition.

Pat knew that 3 + 4 = 7.

Then Pat asked, "What would happen to the equation if I added
1 to both sides?" Pat tried it and got

3 + 4 + 1 = 7 + 1

Pat realized right away that this new equation was also true.
Then Pat went back to the original equation of 3 + 4 = 7, decided to
subtract 3 from both sides, and got

3 + 4 - 3 = 7 - 3

Pat then did some arithmetic and ended up with 4 = 4.

Right away, Pat realized that this technique could be applied to
different types of equations. Pat asked, "What if I didn't know one of
the numbers?"

Pat was already familiar with equations like 3 + 4 = ? and knew
that you could solve those equations.

Pat decided to try something a little different: ? - 4 = 7. Pat knew
from before that you can add or subtract the same number from both
sides of an equation (see above) and still have a true equation. So
Pat added 4 to both sides of this equation and got

? - 4 + 4 = 7 + 4

After a little bit more arithmetic, Pat ended up with ? = 11. If you
keep thinking like this, and instead of using ? you use x or y or a to
stand for the missing number, that means that you are starting to
think algebraically.
-Dr. Math, The Math Forum

Chapter Two

A variable is a symbol like x or a that stands for an unknown quantity
in a mathematical expression or equation. If you remember that
the word variable means changeable, then it is a little easier
to
remember that the value of the x or a changes depending on the
situation.

For example, what if you are thinking about the number of tires
you need for a certain number of cars? You know that 4 tires are
needed for each car. You can write 4c = t, where c is the number of
cars, t is the number of tires, and 4c means 4 times c. If there are 25
cars, you can figure out that 4(25) = 100, so you will need 100 tires. If
there are 117 cars, you know that 4(117) = 468 and you will need 468
tires. Because the number of cars can change but the relationship
between the cars and tires stays the same, the formula 4c = t is a
useful
way to explain the general situation.

By the way, in an expression like 4c - 3, 3 is called a constant,
because it doesn't vary. The 4 changes along with the variable it
multiplies and is called the coefficient of c.

What Are
Variables
For?

Dear Dr. Math

Why is it important to be able to figure out
the values of variables? We've been doing
the in our math class for more than half a
year and I was just wondering why we are
doing it.
-Carissa

Hi, Carissa,

This is a very perceptive question.

Variables are important for a couple of reasons, which we might
call planning and analysis.

Think about planning a dinner party. Let's say you know that
you'll need one-half of a chicken for each adult and one-quarter of a
chicken for each child; you'll need one bottle of wine for every three
adults and one bottle of soda for every five children; you'll need a half
pound of potatoes for each chicken that you have to cook; you'll need
one pie for six adults and one bowl of ice cream for each child; and
so on.

But you don't yet know how many people you're going to invite.
Variables let you set up a description of the situation (i.e., an equation)
such that you can plug in two numbers (the number of adults
and the number of children) and get back other numbers that you'll
find useful: how many chickens to buy, what the total cost will be,
and so on. If you decide at the last minute that you want to add three
more guests, you don't have to start your calculations from scratch-you
just change the values coming in and the equations will tell you
how to change the values at the other end.

This, by the way, is why they are called variables-they tell
you how some quantities vary in response to changes in other quantities.

Note that a dinner party isn't all that complicated, so it's almost
not worth the effort of setting up equations to solve the problems. But
when you get to something more complicated-like trying to plan
the flight of an airplane or run an entire airline-it becomes
absolutely necessary to use variables. A big part of running any
business is being able to figure out your potential costs in any situation,
because that tells you how much you need to charge for goods
and services in order to make enough money to stay in business.

So, that's planning. What about analysis? Well, analysis is just
planning in reverse. If you know how many people to invite, you can
figure out how much money you'll have to spend. That's planning. If
you know how much money you spent, you can figure out how many
people you invited. That's analysis. The beauty of variables is that
in most cases you can use the same equations to go in either direction-to
predict what's going to happen or to understand what
already happened.

The planning aspect tends to be more useful in things like business
or construction or engineering, where you have to decide what's
going to happen. The analysis aspect tends to be more useful in
science, where you don't get to decide what happens (the world
behaves the way it behaves, whether you like it or not) but would
like to understand it anyway, whether due to curiosity or because
you'd like to use that understanding to make your planning more
accurate.
-Dr. Math, The Math Forum

Using
Variables

Dear Dr. Math,

How come when you use a variable in a
problem sometimes the answer still has a
variable and you cannot get an actual number
answer?

Thanks,
-Clive

Dear Clive,

I assume you are not talking about making a mistake in solving the
problem. If there is only one variable in the original equation, then
either you can solve it with a numerical answer or you simply can't
solve it-there would be no actual solution that still involved the
variable.

But if you are given an equation with two variables in it, like
w = 24/h, and are told to solve it for one of the variables, say h = 24/w, then
the other variable will still be there. In this case, you are simply
rearranging a formula for a different use. As given, the formula lets
you get the width of a rectangle given its height. After you solve for
h, it lets you find the height of a rectangle given its width. You don't
know either one yet, but if I gave you a height, you could plug it right
into this formula. If you hadn't already solved for h, you would have
to put my value into the original equation for w and then solve that
for h.

So, there are two ways a variable can be used. Sometimes it is
an unknown, which you want to figure out from the equation. Other
times it just stands for a value that you don't know now but will know
later, like w in my example. Then you just work with it as if it were
a value but without being able to do the calculations. When you're
done, you can replace it with any value.
-Dr. Math, The Math Forum

Writing
Expressions
with
Variables

Dear Dr. Math,

Here's a problem that I'm having trouble
with:

Write an expression that represents a
$500 donation plus $5 for every event.
Let n represent the number of events.

I do not understand what the problem wants
me to write.

Yours truly,
Clive

Hi, Clive,

An expression is a collection of numbers and variables connected by
arithmetic operations (add, multiply, etc.), so if you worked out all the
arithmetic (which is called evaluating the expression), you would
get a number. In this case, the number would be the total payment.

Let's say you knew there were 4 events. Could you then work out
how much to pay? It would be

500 + 5 · 4

in dollars. This is an expression. You can work it out and get the
answer 520.

In fact, you don't know how many events there are. But whatever
that number turned out to be, you could put it in place of the 4 in the
expression and work it out in just the same way. So we use a letter
as a name to stand for whatever number we will end up putting
there. This is a variable. It is sort of a placeholder for a number.

You were asked to use the letter n to represent the number of
events. That means n will be our variable and we can put it in place
of the 4, like this:

500 + 5 · n

When multiplying by a variable, you don't need to write the "·". You
can just write

500 + 5n

I hope this helps you work out other problems in writing expressions.
-Dr. Math, The Math Forum

Understanding
Variables

Dear Dr. Math,

I have always had a hard time with algebra.
It makes no sense to me why someone would
replace a number with a letter. Is there
some secret pattern that could help me solve
algebra problems?
-Carissa

Hi, Carissa,

You're doing something that's a little like algebra whenever you use
a pronoun. You could have written this:

Carissa has always had a hard time with algebra. It makes no
sense to Carissa why someone would replace a number with
a letter. Is there some secret pattern that could help Carissa
solve algebra problems?

When you wrote to us, you used the pronouns I and me to take
the
place of your name. A variable is like a pronoun: it's a way of talking
about a number without calling it by name.

We don't generally replace a number with a letter; more often
we don't know the number yet, so we just give it a nickname (like x)
and work with it until we can replace the letter with the right
number.

The great discovery that made algebra possible was the realization
that even if you don't know what a number is, you can still talk
about it and know certain things about its behavior; for example,
no matter what the number is, if you add 2 to it and then subtract
2 from the result, you'll have the same number you started with. We
can say

x + 2 - 2 = x for any x

Here's one secret that may help you: when you see an equation
that confuses you, try putting an actual number in place of the variable
and see if it makes sense. For example, in what I just wrote, you
could try replacing x with 47:

47 + 2 - 2 = 47

It works! Now think about why it works: 47 plus 2 means you've gone
2 units to the right; minus 2 takes you 2 units back to where you
started. It doesn't matter that the place at which you started was 47;
adding 2 and subtracting 2 undo one another.

I'll take you one step deeper into algebra and actually solve an
equation. Let's say we're told that

3x - 2 = 7

In words, I can say, "I have a secret number. If I multiply it by 3 and
then subtract 2, I get 7. What is it?" (Notice how I used pronouns to
stand for the number.) In order to solve this, I can think of it as if the
x were a present someone wrapped up for me. First, someone put on
some "times 3" paper and then over that some "-2" paper. The pack-age
I was given is a 7. I want to unwrap it and see what the x is that's
inside.

To take off the "-2," I can add 2 (remember what we said before
about adding and subtracting 2 to both sides of the equation). It
works like this:

3x - 2 = 7
3x - 2 + 2 = 7 + 2
3x = 9

So I've taken off the "-2" paper and what I found inside was a 9. Now
we can take off the "times 3" by dividing both sides of the equation
by 3:

3x ÷3 = 9 ÷ 3
x = 3

Now the present is unwrapped and we can see what it is. We
were able to do all this because we knew how to handle a number
without knowing what it was. Of course, since we can always make
mistakes, we should check that we're right; let's wrap it back up and
see if it's a 7:

3(3) - 2 = 9 - 2 = 7

Yup! That's what was in the package. And doing this lets us see what
was happening to x by putting a real number (the right one) in its
place.


Continues...

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Excerpted from Dr. Math Gets You Ready for Algebra

Copyright © 2003 by The Math Forum.
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