AdamJTP

For me maths isn’t a spectator sport. It’s a game to play, even if you don’t play it well, and I try to teach my children this too. 

This is what we did after Saturday lunch. I’ve put this here for my children to refer back to, and in case anybody else finds it useful.

On Saturday I was talking to my children about population dynamics.

We started talking about populations that doubled in number after each generation:
1,2,4,8,16,32,64,128,256,512,1024 …

Then Fibonacci’s rabbits:
0,1,1,2,3,5,8,13,21,34,55,89,144,…

…and we all agreed that if rabbits really did breed like this we’d be up to our armpits in rabbits by now.

So I asked them to grab their laptops and fire up Excel.

This is a famous (and simple) population model.  All you need to know is the number of "rabbits" will vary between 0 (no rabbits) and 1 (too many rabbits i.e. complete environmental destruction by rabbits!).

If you’re a reading sort of person then Wikipedia and the web will explain it better than I can – http://en.wikipedia.org/wiki/Logistic_map

Whereas here are the "just play with it" instructions for eight year olds. 

There are applets that do this on the web – but I wanted my nippers to create the spreadsheet themselves.

Creating the experiment

1. Open Excel
2. In cell A1 type the number 2

3. In cell B1 type the number 0.01 (this is the starting population. Remember 0 is none, 1 is too many).

4. In cell B2 type “=A$1*B1*(1-B1)”

4a. We talked a bit about what that equation meant – but anyway, who cares…

5. Click on cell B2, grab the little square on the corner and drag it down for at least 100 cells.

6. Let go and scroll back up to the top

7. Click on the B at the top of the page to highlight all of column B

8. From the insert menu, choose line chart

9. Drag the corners to make the chart bigger

Now – playing with the numbers

See that number 2? That’s our “breeding factor”, change it to numbers between 1 and 4.  This is the whole point of the experiment to be able to change these numbers ourselves.

Cell A1 = 2

Gives us the graph below – the graph goes up and stabilizes at 0.5.  My children seemed to think this was the “right amount” (presumably because it’s between 0 and 1).

Cell A1 = 2.6

Ooo unexpected! This overshoots a bit before stabilizing at 0.615.  The children already like this model – the number of rabbits is limited (we seem stuck with the rabbit metaphor – damn you Fibonacci!)

Cell A1 = 3

Hmmm… that’s tricky, the number bounces about a bit.  Maybe it would get down to a single value if left long enough.

Cell A1 = 3.2

Oh no – there’s too many rabbits then the population crashes (then presumably the grass grows back) and then loads of rabbits again

 

Cell A1 = 3.5

High, low, not so high, not so low, high, low, not so high, … funny looking graph.

3.8

Now it gets interesting.  The graph is all over the place!  It swings about, seems to find a level then it’s off again.

but what about….

3.8000000001

We’ve changed the breeding factor by a tenth of one billionth and got a different graph.  Okay, they’re both spiky but it is clear that the difference is more than a tenth of a billionth.

Further investigation

This is then a good point to talk about the butterfly effect (how a small change in a number lead to a big change in the graph).

And then I left them exploring different numbers. However, navigating is always easier with a map.  If you’re trying this yourself – see if you can work out how the map below relates to the graphs above.

Or here’s a video from the excellent sixty symbols website.

…and then…

…we ended up setting up a model to explore the likelihood of “assembling a 747 in a junkyard using a tornado” using a set of wargaming dice.  Okay, we didn’t make a 747 (using wargaming dice?  C’mon that would be weird!), but we did beat godzillions-to-one odds.  But I’ll blog about that another day (unless we find a dead rabbit to mummify first – because that would be VERY cool).