A Four-year-old Thinks About Even and Odd Numbers

Date: 07/22/99 at 03:02:40
From: Timothy Pfister
Subject: Even numbers

(I'm typing this for my son, in HIS words, as he gets frustrated with 
"writing" with the keyboard. He's pre-K, but please answer if you can 
- I'm a housewife, not a mathematician. Some of this was asked to me 
personally, with a little extra banter between. I've done my best to 
translate.)

"Mom told me that even numbers are ones that can be cut in half and 
both parts are the same size - like when I have to eat lunch and she 
has to go to the store, and she only makes me finish "half" of it. 
"Half" means 2 parts, and some numbers are not that way.

Mom says they're "odd." It's not fair letting 2 be even all by itself! 
What about 3? I thought about 4, but that's just 2's twins, so it's 
okay. "3" has 3 parts that make the whole thing with no bits, and when 
we counted 9, I found out it has 3 parts made of 3. Mom says that 
"odd" numbers have bits left over, so why are some numbers called 
"odd" when they don't have bits? Mommy says "odd" numbers aren't 
"weird." It means something LEFT OVER, but I don't think she knows 
because she isn't a teacher.

FOOD isn't made of numbers unless it's a "PI." When I play a game with 
Adam (my brother) and we're even, nobody gets mad and we both win. But 
when David (my other brother) says I'm "the odd-man-out" I get mad 
because he won't let me play then.

I'm going to school when I'm bigger, and I have to know all about 
numbers so I can be smart. Mommy says 5 is an "odd" number, but it's 
bigger, and besides, it has to use 2 and 3 and 4 like step-stones to 
get to 20. When I'm 5, I won't cheat like it, because I have to go 
(counting from 6-19) to get to be old!..."

(It goes on, but I think you get the gist...HELP!)

Date: 07/22/99 at 09:10:15
From: Doctor Peterson
Subject: Re: Even numbers

Hi, Timothy! I don't think I've ever written to such a thoughtful 
four-year old. (My youngest son will turn five next week, and my 
oldest son is a Timothy, and I love talking with all my kids about 
math!)

It sounds like your Mom has told you many of the same things I've 
taught my kids about even numbers. We call a number even if it can be 
paired off without any extras; 5 is odd and 6 is even because they 
look like this:

    * * *       * * *
    * *         * * *
      5           6

If you talk about cutting numbers in half, you have to be careful to 
remember that the two "halves" have to be the same size, and that you 
aren't allowed to cut a pea in half to make it even. Once you talk 
about fractions (pieces of things), the idea of evenness no longer 
means anything. That's why I like to put beans or marbles or something 
in rows when I talk about even numbers.

When you pair off people, such as to play games with two teams, the 
one left out is the "odd" (left over) one, which is why we also use 
the word "odd" for something different that doesn't fit in. It's fun 
thinking about how different meanings of words fit together.

The number 9 is odd because it doesn't pair off evenly:

    * * * * *
    * * * *
        9

But it does "triple off" evenly:

    * * *
    * * *
    * * *
      9

We say that 9 is divisible by 3, but not by 2. We don't have a word 
for numbers that are divisible by 3, as we do for numbers that are 
divisible by 2 (even). Whenever we talk about even and odd, we're 
always thinking of twos. But dividing things by other numbers can be 
very important. Someday you may learn about "modular arithmetic," and 
you'll see that it can be very useful. And before then you'll learn 
about prime numbers, which can't be divided evenly by anything. Maybe 
you can have fun by trying to find which numbers are prime - for 
instance, try to arrange 11 beans into rows, and you won't be able to 
do it (unless you call it 1 row of 11 or 11 rows of 1). Prime numbers 
are REALLY "odd."

I'd love to hear from you again. Numbers and words are fun to talk 
about.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/