Learning Addition and Multiplication with Algebra

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

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 
one apple and completely forget about it, that brings your total 
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, 
please write back.

-Doctor Aaron,  The Math Forum

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

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 
talking about.  It's what algebra is all about: using letters to 
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.

Now, about your other question.  There is some connection between 
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