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### Variables: Connecting Letters and Numbers

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Date: 12/06/2001 at 12:48:12
From: Elizabeth
Subject: Understanding the connection between letters and numbers in
algebra

No matter how many books I read, I can't understand how x = whatever.
I'm still trying to know how you make a connection. Is there some
sort of table you can use to show what letter is equal to what number?
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Date: 12/06/2001 at 14:10:56
From: Doctor Ian
Subject: Re: Understanding the connection between letters and numbers
in algebra

Hi Elizabeth,

A variable is just a name you use to talk about a number whose value
you don't know yet.

Suppose I tell you that if you add John's age (in years) to Joan's age
(in years), you get 27, and if you multiply their ages, you get 140.

Now, you know that John _has_ an age. You just don't know what it is.
And you know that Joan has an age, but again, you don't know what it
is.

However, you know some facts about their ages, and in order to express
those facts, you need some way to refer to their ages. You could write
something like

John's age + Joan's age = 27

John's age * Joan's age = 140

Now, this is more writing than you probably want to do, so you might
choose shorter names.  For example, you might use 'J' to stand for
John's age, since the J would remind you of 'John'. But this means you
can't use 'J' for Joan's age, so you need to think up another name.  A
somewhat natural choice would be the next letter of the alphabet, 'K'.
Now we can write

J + K = 27

J * K = 140

At this point, it's important to realize that these are just arbitrary
names. And a couple of important points follow from that:

1) We could use any other names without changing the meaning of the
equations. For example, we could use 'X' instead of 'J':

X + K = 27

X + K = 140

And we could use 'Y' instead of 'K':

X + Y = 27

X * Y = 140

The particular variable names that we use don't make any difference,
so long as within a given problem, the same variable name always has
the same meaning. (It's sort of like in a novel. It doesn't matter
what name is given to any character, so long as each name stays with
the same character throughout the book.)

For that matter, you don't even have to use letters for names. You
could use pictures if you wanted to. But letters have a few nice
things going for them. For one, most people can recognize them, and
can agree on how they should be pronounced. Also, most people already
know how to write them, or how to type them with a keyboard. Also,
they aren't used for operations (like addition or multiplication), so
if you use a letter in an equation, people reading the equation can
have a lot of confidence that it is a variable name.

2) The value that turns out to be associated with a particular
variable name can be different from problem to problem. For example,
in this problem, it turns out that J and K have the values 20 and 7:

20 + 7 = 27

20 * 7 = 140

But here is another problem that uses the same variable names:

John has twice as many cookies as Joan. If he gives her two

Let J stand for the number of cookies that John has, and let
K stand for the number of cookies that Joan has.

J = 2K          John has twice as many cookies

J - 2 = K + 2       A transfer of two cookies makes them equal.

In this case, it turns out that J has the value 8, and K has the
value 4:
8 = 2 * 4

8 - 2 = 4 + 2

So there is no table, or other device, that you can use to simply
'look up' the value of a variable based on the name that has been
given to it. Is J equal to 29, or 7, or 8?  It depends on the
particular problem being solved.

(Again, it's sort of like with books. If you see the name 'Chris'
being used as a character in one book, and you see the same name being
used in another book, you don't assume that they refer to the same
character. And you don't know anything about the character until

Because there is no connection between the name of a variable and the
value that it takes, it's common to just use the same names over and
over - x, y, and z, for example; or n, or k; or a, b, and c.

As a way of getting used to the idea of variables, you might consider
using entire names instead of letters. For example:

A garden is 6 feet longer than it is wide. The perimeter of the
garden is 30 feet. What are the dimensions of the garden?

1. Write an expression for the perimeter:

30 = perimeter

= length + width + length + width

= 2*length + 2*width

= 2(length + width)

2. Write an expression for length in terms of width:

length - width = 6

length = 6 + width

3. Substitute and solve

30 = 2((6 + width) + width)

= 2(6 + 2*width)

= 12 + 4*width

18 = 4*width

18/4 = width

Now, when you see something like this, you don't get the feeling that
you can just 'look up' the value of 'width', do you?  It's pretty
clear that the width could be any value, and your job is to find out
what value makes the equations true.

Once you get used to working with words as variable names, you'll get
more comfortable with the idea of variables in general; and eventually
you'll decide, in your own time, to start using letters instead of
whole words.  It's a little like learning to stand up before learning
to walk.

You might also find it useful to read these answers from the Dr. Math
archives:

What is Algebra?
http://mathforum.org/dr.math/problems/jason.07.20.01.html

Letters for Variables
http://mathforum.org/dr.math/problems/spencer.11.19.01.html

I hope this helps.  Write back if you'd like to talk more

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
Middle School Algebra

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