Monday, 9 October 2017

Exploding Dots - for Fraction Division (operations)

So... I came across this tweet earlier tonight




While I have typically leaned on this resource from Marian Small, I was curious about using exploding dots for something like this.

If you are not aware, this week is #GlobalMathWeek, lead by Global Math Project's work with exploding dots.  I highly recommend signing up if you have no already.

The idea of exploding dots leverages the deep concept and structure of place value, and use it to basically explode your mind :)

In any case, let me get on to doing what I came here to write about.

Fraction division.  The dreaded topic of many.  Again, there are many great resources out there using area and number line model.  Here, I am mainly playing with the exploding dot ideas.  I dug around for existing work using exploding dots for fraction division, but I came up short.  So I thought I'd play around with it and see what comes out.

Before I begin, I'd like to highlight that I am just playing.  I am unsure if it completely follows the logic that has been previous established by James Tanton, although I certainly tried hard to preserve the concepts/structures there.

I'm going to begin with the idea of representing 3/2 with a box.  Note that for my purposes, I only need two boxes.


As you can see, when the box on the right fills up, it becomes a '1' which goes in the left.  So I can use a similar idea to 'un'explode the dots to make 3/2

Ok so now I move onto something like (3/2) times (2)

Well that's just taking the dots and then duplicating each one.  All at the same time, remaining in the (1/2) box, which means I have 6, as seen below



Which can, in turn, also turn into 3 groups of 2 that 'explodes' into the 1's.  In other words, equals 3.

But wait, I'm not doing multiplication here.  So let me move into playing with division.

So (3/2) divide by (2) is what I want to do next:



So here is where it gets tricky.  In order to 'divide' by two, I am splitting the boxes into two.  Each of my 'sub'boxes now need to be filled before the entire box is filled and I can explode it into the 1 box.

And so since I now require '4' dots to fully explode, I have (3/4)

Ok so that was alright.  Let me do (3/2) divide by (3)

In the same vein, I split the existing box into 3 pieces


Each 'sub'boxes require 2 dots filled in order for the whole box to explode.

And so since I need 6 dots in total to explode, and I only have 3, I end up with 3/6.

Ok so now I need to add a new idea of what happens when I divide a fraction by a fraction....

(3/2) divide (1/2)

In this case, I split each dot into two.  But each half is now counted as a full one (since we're counting by halves).  So I can think of it like mitosis or the fact that I am counting differently now.  This is similar to, but slightly different than, how I multiplied by 2 earlier.



And so once I count by halves, I note that I now have 6 dots in that box, where every 2 can explode into the next one.  And so, once again, I end up with 3.

So putting everything together...

If I have (3/2) divide by (3/2):


So here, I begin by both splitting the dots into 2 (doing the 1/2 aspect), as well as splitting the boxes into 3 (doing the divide by 3 aspect), which then leaves me with 6 dots but in 3 'sub'boxes.  Since now I have each box being full, it fulfills my requirement of the entire box being full, and so I get 1.




As I mentioned in the beginning... this isn't really polished, and as I mentioned here, it isn't as intuitive as I would like it...




And so this is more of an open invitation for others to think about this with me!

Let me know your thoughts.

Let's have lots of fun exploding dots this week!






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