3-digit numbers and a Dienes-hundred square combo

...And while I've got a moment, another thing from Friday.

I mentioned just now in the last post (two coming along at once!) that one of the things on Mike Flynn's list of generalisations is:

• Our number system is organised by powers of ten (base ten).

We'd talked a lot about ten, explored it in all sorts of ways. Presented an assembly about it. Made a book about it:

Open publication 

But the way ten figures in organising our number system hadn't come up. I got a couple of my students to make this staircase, with just minimal help:

and looked at it a few times on separate occasions:

EH's question, Why are there a lot of oranges? - and the fact that no-one had any response to it, made me feel that there was no real feeling yet of the ten-ness of our numbers. I resisted the temptation to say anything in answer.

I'd read Alexandra Fradkin's post, 3-digit numbers are tricky! Part II, and I've been pondering it.

Our number system is, as you know, just one of many. The Romans system I, II, III, IV, V, VI... carried on in Europe until and beyond the time that Fibonacci introduced the Hindu-Arabic system that we use. There is a lot of logic to our system, but it's also one human-created system among many possible ones.

So there's no reason children should be able to anticipate how it works, without orientation of some kind. But then again, there is such a powerful logic to it, that, given the orientation we're given, we can all write numbers we've never seen before.

My question was, what is the minimum amount of orientation you can be given to learn about this, so that you do the maximum amount of seeing logic? Well, actually it was more particular: what was the minimum amount of orientation and maximum amount of seeing logic my class needed? My class with whatever contingent things happened to come up.

I've approached it Alexandra's way before. It's quite impactful to get it wrong, to see your first assumptions weren't correct. I also think it's enjoyable to get things right.

What did come up was, when I asked the class to bring in something about ten, BG brought in this:

The idea of 2-digit numbers seemed to me to be a key that could unlock it all.

When we'd looked at BG's first two sentences carefully, I asked the class for some other two digit numbers (in red below). Then I said, what about one-digit numbers? It was someone else who said, what about three-digit numbers? 100 was the first one to come. I can't remember how the others came.

This seemed like the moment to get the little whiteboards out and ask everyone to cross that threshold from two- to three-digit. Someone had proposed a hundred-and-one as 101, so it was going to be manageable for everyone.

There was a good feeling about this, like we'd just crossed a new boundary to explore new territory together.

But though we'd crossed this river, there was no reason to think that meant the base ten system was instantly understood. How to help with that? What would you do here?

We'd looked at the hundred square lots. I use this online one.

I thought of Dienes ("Base ten") apparatus, which embodies our number system so well.

And wondered if we could combine the two, to give a sense of the economy of our number system.

I asked the class in pairs to stick a hundred square behind our hundred square frames, and then to fill up the square up to a particular number. 

I know; this is a bit of a “I, We, You” lesson, rather than a  “You, Y’all, We” one. It was an experience that I thought would help to throw into the pot, for seeing numbers in terms of tens and ones. But maybe - no, I'm sure - there's a better way. What are your ideas about this?

I feel like with all our work on making sense of numbers, and now that the class are more independent in their writing, it's maybe time to start a kind of journal entry on this, something like 'What I know about numbers. What questions I have.'

I'm interested what they would say, whether this will be too big a challenge at this stage...