Let’s explore patterns

Welcome back mathematicians, and I'm here with one of my favourites today, hi Sam.

Hi Michelle. How are you?

Good. Very good. So I've got this box here of equipment today.

Ok.

And what I wondered is if you could sort them for me. Alright, maybe you could write down the different types so it would it be easier. So as I sort them out you could write them then we could put their little sections.

So do you think there's going to be more than one way to sort them? Yeah, OK. Let's have a look.

Do you want to tip them out of the box or do you want to keep them in there? We could tip them out. Alright, what are you thinking first Sam?

I'm thinking things that might roll. OK. So that definitely rolls, so we could put that over there. That would probably roll, yep, and all these are wheels too, so they would roll and this little car. So that's like the wheel section I guess. There may be like almost animal stuff.

So would you say that these are things that roll and things that don't roll? Yeah.

Would they be our categories? So things that roll and things that don't roll.

OK, and then what else are you thinking about?

If you put them all back together. Maybe things... Hmm... We could do also like almost mostly living things because I guess you could call these living things and then non-living things.

Ok. Living things and non-living things. Ok, yep, what else could you do? What's another sort you could do?

You could also do things that stick together like these and stuff and then these won't stick together or anything.

So I'll move those over here then to help. These don't stick together almost, so I guess you use those, but I guess you could say that sticks together because it's made of stuff. But would it stick to another car? That's a really good question. Probably not. So, but this would stick together because like you can take it off then put it back on. Oh, not probably not.

So, yes, all these, oh no probably not, so all these things stick together onto stuff. Oh, yeah, I see. Then you could, I guess you could make smooth things and rough things. So that's smooth, that's smooth.

Hey, I've got an idea Sam. This is going to be stretchy for the brain. If you come back to this category where you've got things that stick together and things that don't stick together. So these are things that stick together and these are things that don't stick together. I wonder if now you could do a second category inside that so things that are smooth and things that are not smooth and you could then say...

Well, it's a bit tricky with all the Lego because they all have smooth things on that. Maybe you could say something like things that have that have more colours than black-and-white. So for example, this shark and the car could then move down here, and maybe you could say, these things 'cause I have grey, but these things are all black and white, whereas then all those things would be down here. So you'd have like if you went this way, you're saying things that have more than black and white, things that have color.

I guess that could go there 'cause it's bright yellow. Yeah, but it's also over here. 'cause it's things that don't stick together. Does that make sense? So come up with another category. I'll show you again.

What's another way you could sort them? Length? Try.

I guess you could say it would have to be bigger than this one.

Yeah, so so that's definitely bigger than the car and the wheel...

No, I, so that would go over there and all these are definitely smaller.

So I guess all of these would go over there too. The shark's bigger so that would go over there.

That's bigger... That's definitely smaller. We know that's all smaller because we just measured that. All these are smaller then.

Smaller. Smaller, smaller, smaller, smaller.

Yeah so. All of these are smaller. All of these are bigger. Is there another way?

Yeah I guess then from there you could also do the same like with colour.

Yeah do you want to try that? It's quite tricky because you're then looking at instead of just two categories, but a two by two sort. So let's move this over this way so we can see it like this like this and over here.

So Sam almost, what's happening is, almost like you've got like a grid now. Don't you? Where you have got things that are black and white. And then things that are coloured. And then you've got things that are smaller than the polar cub. And bigger than a polar cub. Does that make sense?

Yeah, exactly right you so you could do see how they would fit in like that. Does that make sense? Then Sam, so now you've got a two by two sort. So you started off, so you could say here these are things that are colored. And these are things that are black and white. And then you can also say these are things that are smaller than the polar bear, and these are things that are bigger than the polar cub. Ah, that's an interesting way to sort.

I wonder if there are other ways that you could sort them. Yeh. That's a good question.

Hello Sarah. Hi Michelle, how are you today?

I'm very well how are you?

Very good thank you.

We thought that we would come together today to explain some of the complexities around mathematical patterning. It seems like a really simple thing to notice and wonder about patterns, but sometimes little kids find this really tricky and it can be quite difficult to think about how to help them through identifying and noticing and extending and exploring patterns.

So we thought we'd come with some tips and tricks for teachers in classrooms and mums and dads and family at home.

Sounds great. Thanks for helping me.

So, Sarah, I've prepared a pattern underneath my sheet of paper and I wonder at the moment if you can workout what

my pattern is. If I said to you what block comes next, can you answer that question yet?

Not confidently.

So you'd need some more information?

I need some more information, yes.

What if I show you my next block, can you tell me what comes next now?

Not yet. I can start speculating, but I still don't really know confidently.

Ok. what if I reveal my next block?

OK, yeah, I'm starting to hypothesise a few things.

So what are you thinking it could be?

I'm thinking that it could be green if it's just a simple, alternating pattern. Yes, that's what I am thinking but I'm very keen to see if potentially there is different colour being introduced, or maybe another yellow.

So, you're thinking it could be green like this?

It could be, yes.

Could it also be something like red?

Definitely.

Could it also be something like another yellow?

Yes.

So at the moment we don't have enough information about the core of the pattern to see?

Yes.

I agree with you. What if I now do this?

OK yeah, I'm starting to get a bit more confident but I still don't fully trust it yet.

And that's actually really wise decision because what we want to see to say something has a mathematical regularity is we want to say that it happens over and over and over. So we've sort of seen yellow, green, yellow, green over and over, but we're waiting to see if we see it maybe for the third time to be more confident that it's definitely there. So you're starting to think it's probably yellow?

Yes, my hunch is yes.

OK and if I reveal again now we have green and now you're like yeah this is the pattern and so we could describe this as a yellow, green, yellow, green, yellow, green and what would come next then?

Yellow.

Yellow and we have now seen that three times so he's sort of one chunk of the core, here is another representation of the pattern core and here is a third representation of the pattern core.

So that's why you feel a bit more confident, actually, to say I think this is now going to continue, my pattern will continue. Yes.

And so sometimes when we talk about patterns like this with kids, it's important to acknowledge that it's yellow, green, yellow, green and the attribute that's changing here is the colour. It's also important to talk about sometimes people refer to this as a one two pattern.

Sometimes that can be really tricky for students because we'd say one, two, and then we call this one again, which is really confusing when they're young and learning about attributes of counting principles, and so mathematicians would refer to this as an AB pattern. So I say AB AB, AB AB and that way it becomes generalisable so I could then say and build another pattern underneath where I say red, yellow, red, yellow, red, yellow, red, yellow. And even though the colour looks really different, they're actually the same pattern structure because if I squish them together I can see that in fact the core has two things in it before it repeats, but if I call it yellow, green it then makes it really hard for me to generalise out into a situation. Whereas if I say AB, I can still say it applies.

This is why we typically refer to them as AB patterns.

We can also think about this idea of moving from an AB structure like this into another AB structure as the idea is translating a pattern into something different. So this one is not too tricky because we're still working with blocks, but we could also start to do something with equipment at home, like a train and a train track, a train and a train track, a train and a train track and we can talk about that actually this is still an AB pattern, and again, if I make these spaces and do the chunking, that helps me see that in actual fact it is the same pattern core, it's now just represented in an entirely different way. You can also do it with, body sounds. OK. So you could do like a clap and a stomp on the ground, a clap and a stomp, a clap and a stomp. So...a simple thing that you can do is this, ask is how else could we represent the same pattern call using different equipment or sounds or movements in their bodies. I think kids would love that! They could go and find some of this other materials that they could use to replicate the same pattern. Why don't you try to make another AB pattern using this equipment? OK, I will.

Oh, I see, because it's got AB AB AB like our other AB patterns. You could even do something with movement too. So if you change these ones for red. You could do it like this up, down, up, down, up, down. So that the kids aren't always just having to think about colour, but they're looking at other attributes, and in this case their position.

Yeah. There's another way to represent it, vertical, horizontally, vertically, horizontally, vertically, horizontally, still AB AB AB, and so some of the questions that we can ask of students are things like - Now what would be next in my pattern and ask them to continue it.

So what would come next in your pattern that you are making?

It would be a vertical and a horizontal.

And then we can do things like saying - uh oh, something came out of my pattern, what might that now be? OK, so you would then sort of cover it or get them to close their eyes, is that what you're thinking?

Yes. And so this is much trickier for kids to see because the pattern core is disrupted, and so that's where it might be worth saying - Oh well, let's see if we aren't sure what comes here, let's separate the chunks that we can see like this. Yes, OK. And in actual fact, then I could move them and say they're exactly the same, and they're exactly the same and if I move this here, now I can see what's missing. Yes. I can put the piece in and then I can re-organise and re establish my pattern.

Brilliant. Have fun patterning mathematicians.