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### Learning Addition and Multiplication with Algebra

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Date: 3/16/96 at 20:5:52
From: Anonymous
Subject: The (a) in algebra

Dear Doctor Math,

I am 8 years old. How do you know what the (a) is in algebra? I saw
some algebra on a picture. I like math, so send me a good letter
so that an 8-year-old can understand. My name is Jeremiah.

Jeremiah
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Date: 3/19/96 at 14:23:39
From: Doctor Aaron
Subject: Re: The (a) in algebra

Hi Jeremiah,

I like math too.

I'm taking a guess here, but the picture you saw probably looked
something like:

5 + a = 12   or    6 x a = 18.

Just think of the (a) as a (?).  Then if the equation is:

5 + a = 12 or 5 + ? = 12, we have to figure out what number a or ?
could be to make the equation true.  Maybe you can just look at
the equation and see the answer, but if you can't, there are
tricks you can use to solve the problem anyway.  Since we know
that adding 5 to (a) gets us 12, and that adding 5 is just
counting upward from a by 5, we can count backward from 12 by 5 to
get back to (a).

It might help to think of a real world example.

Let's say that you have 12 apples (if you're bored of fruit in
math use 12 hamsters).

Now you're going to close your eyes and someone else is going to
put some of the apples into a black box so you can't see them.
You can count the apples that aren't inside of the box.  You count
5 apples.

Well, you know that you started with 12 apples.  If you take away
number of apples down to 11, the apples outside of box down to 4,
and you haven't touched the apples inside of the box.

If you take another apple away, your total number is down to 10,
you have 3 apples that you can see, and you still haven't touched
the box.

We could keep doing this until you get down to zero apples that
you can see and all of them are in the box. Then your total number
has to be the same as the number of apples in the box, so we know
what a is.  This could take a while if I asked you to solve 25 + a
= 27.  That's why it's a good thing that somebody thought of
subtraction.  When we keep taking apples away, we're really just
subtracting the apples that we can see from the total number.

So if  5 + a = 12, then 5 - 5 + a = 12 - 5, but 5 - 5 = 0, so

a = 12 - 5 = 7.

If you understood all of that there are some more key concepts
that I don't want to leave out.

(a) can be in more that one equation, and stand for different
numbers.  This is really important.  If I ask you to find (a) when
4 + a = 7, and then I ask you to find (a) when 4 + a = 9, you are
going to get two different answers for (a).  This doesn't mean
that you did one of them wrong, or that (a) is both 3 and 5, it
just means that (a) only makes sense with the problem that you're
working on at the time.  If we think about the black boxes again,
it would be silly to think that every time you closed your eyes
and someone put some apples into a black box, it would always be
the same number of apples.  Well, since the black box was just our
real-world way of thinking about (a), (a) can also be different
depending on the problem that we're working on.

The last thing I'm going to talk about is multiplication - because
its a little trickier than addition.

Let's say that 6 x a = 18.  This is like saying that we have 18
apples.
When we close our eyes, someone comes and puts the apples into 6
separate black boxes each with the same number of apples.  Another
way of saying this is that 18 apples are divided equally among 6
boxes.

Good thing we have division to tell us how many apples are in each
box.

Then 18 / 6 = 3 is the number of apples in each box.

I'm going to stop here.  If you want me to give a better
explanation of algebra with addition and multiplication, or if you
understand it and want me to talk about more complicated algebra,

-Doctor Aaron,  The Math Forum
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Date: 3/29/96 at 15:50:19
From: Anonymous
Subject: Thanks for the letter

Dear Doctor Math,

My name is Jeremiah. You sent me some info on "the (a) in
algebra."

What I said was not like this: 5 + a = 12. This is what I saw:

a(1/a) = 1/a(a) = 1.

I don't understand that. Can you send me some good info? I'm 8
years old.

I thought it was funny when you said that if I'm getting tired of
using fruit in math, I can use hamsters.

I like chess a lot. Is chess linked with math?

Jeremiah
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Date: 5/7/96 at 13:7:28
From: Doctor Ken
Subject: Re: Thanks for the letter

Hello!

That "a" is essentially the same kind of thing that Aaron was
stand for numbers when we don't know exactly what the numbers are.

The reason we do that is that lots of times when we write down
some equations (like a + 5 = 7   or 3/a = 6) we can actually
figure out what a is.  Incidentally, we don't HAVE to use "a".  We
can use x, or y, or a little balloon shape, or whatever we want
to.

Let me help you translate the thing that you saw.  It said that
a times 1/a equals 1, and 1/a times a equals 1 also (it becomes
customary in algebra to leave out the x symbol for times in most
cases - do you think that's good?).

So in order to understand that, you have to know what the
expression 1/a means.  Do you know what fractions like 1/2 and 1/8
are?  Well, 1/a is exactly the same thing, but now we just don't
know what number a is.  We do know that whenever we see an a,
we'll replace it with some number (a represents the SAME number
everywhere, not 6 in one place and 3 in another place).  We just
don't know what that number is.

So the equation a(1/a) = 1 means that if you take a number, and
you multiply it by 1 over itself, you'll get 1, guaranteed.
Unless that number is 0.  But that's another matter.  Just
remember this: DON'T EVER, EVER DIVIDE ANYTHING BY ZERO!
It'll get you into a lot of trouble.

chess and math.  Most of it has to do with the way people think.
Lots of people say that people who like to play chess also like to
do math, because they like to figure out puzzles and solve things.
There are also some more concrete math things going on in chess.
For instance, number the columns of a chess board 1 through 8, and
number the rows 1 through 8.  Now, put a piece on a black square,
and add the row number to the column number.  What do you get?
What do you get if you're on a white square?  Can you find
something general to say about what kind of numbers you get on
black squares and what kind of numbers you get on white squares?

-Doctor Ken,  The Math Forum

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Associated Topics:
Elementary Multiplication