Division for Total Beginners (or anyone antsy around fractions)

It took me a few days to write this post because, in my brain, division is where simple arithmetic branches off in a dozen different exciting directions and I have to figure out which path to take you down first.

Let's start simple.

Division is the opposite of multiplication.

If multiplication is taking groups and combining them together, division is simply taking a big pile of pieces and separating them out into groups.

To do it, start by counting your pile.  If you're working with a little kid or still getting used to modern styles of math teaching, start by making 10!

Great!  We have 12 in our pile. 

Now, if I tell you: divide your group of 12 into 4 smaller piles, I'm asking you to answer the math question: 

12 ÷ 4 = ???

But I really hate that ÷ division sign because nobody ever uses it after the very first introductions, so with me, you'll usually see it laid out in fractions

12/4 = ???

Awesome!  Just start sorting your pieces into four groups.  Only one important rule: the groups have to be exactly the same.  Tell your kids that, in this game, we must be fair to all the piles.  They all must be equal.  (Kids love putting life in terms of fair and unfair, so this will help the rules of the division game stick).

I like to divide my visual space into four areas (here I use post-its).  It's just visually clearer that way. 

Awesome!  Let your kid try it with a bunch of numbers that are actually divisible by 4.  So... 4, 8, 16, 40, 88.  Whatever.  Start with nice neat easy answers that is streamlined with the bead method of demonstration. 

The problem is, numbers aren't actually beads.  Beads make a lovely model for many situations, but their efficacy is limited.  Numbers are much more interesting and flexible and nuanced than beads.  So our model comes up against a wall when we meet numbers that ARE NOT neatly divisible by 4.  For example, if I add just one bead to my pile of 12, I will have 13.

Let your child try to divide 13 into four groups.  Remember our only rule: it has to be completely fair.  All groups MUST have the same number of beads. 

You're going to come up against a problem.  When all the groups are even, we haven't finished our division.  There's one bead leftover. 

When first introduced to division, children are usually instructed to just call this the remainder. 

13/4 = 3 remainder: 1  (or 3 R:1)

And that's true.  But it isn't totally solved yet.  It's just like we put a placeholder on our process and commit to coming back later when we've learned better tools.  Well, here's the better tool: it's imagination. 

Beads are concrete and awesome, but they are just the beginning.  We need to bust through the limits of this simple model and explore the problem more deeply using our imagination.  

Imagination is FANTASTIC for math thinking, by the way.  Don't ever let anyone convince you that numbers are the opposite of imagination.  Nobody can think deeply in math without a rich, deep, and nuanced creative process. 

Right now, I'm imagining that I can easily slice my glass bead into perfect pieces.  I can't, and I still like models, so I'm going to replace that extra bead with something that I can easily cut up, like a little piece of paper.  

Boom!  Now my glass bead is a paper bead.  No problem.  My brain.  My rules.  The numbers will oblige. 

All my beads are accounted for except for this paper bead.  For this last bead, it's like dividing a cookie evenly among a group of 4 friends.  I just cut it into four equal slices. 

Each slice is called "one forth" or 1/4, and NOW I can distribute the cute little 1/4 fractional pieces into my groups.  My number is divided completely, and I have followed all the rules of fairness in my division game.  

WOO HOO!

How many beads in each pile?  Three whole beads and one forth bead piece.  

13/4 = 3 1/4

The whole process looks like this:

Try again for the next number up: 

14/4 = ?

Well, start with the easy part:

3, Remainder: 2

Ask your child what to do?  He or she might want to make halves.  And that's great!  But I'm going to encourage following our pattern from last time and splitting each remainder bead into 4 equal pieces. Then splitting each remainder bead equally across our 4 piles.  

The very cool thing about this answer is that we get: 

14/4 = 3 2/4

But if you look at those two-forths, you might notice that you can put them together, and once combined, by gum, they make 1/2 the bead!  

Our best, SIMPLIFIED answer, therefore, is three-and-a-half.

14/4 = 3 1/2

To level up the thinking on this, I encourage you to introduce ZERO to your division by sparking this conversation with your child.  

If I have zero beads and divide them evenly on four postits, how many beads on each postit?

0/4 = ? 

(Answer: zero! No beads on any postit.)

However, if I have 4 beads and am tasked with sorting them on zero postits... is there a solution to that problem? 

(There is not! We can not complete the problem and follow the rules of division. The answe is UNDEFINED)

It's a fun brain teaser!