Exploring patterns 1

Welcome back little mathematicians. We're here to talk today about one of the most important things and one of our favourite things about mathematics -patterns. So we've been working on this idea that a pattern is something that repeats over and over and over again. Like these stars - pink, blue, pink, blue, pink, blue. But are these ones a pattern? No, that's right they're all stars, but we can't see anything repeating over and over and over in the colours. What about this picture? Does this have something that repeats over and over and over again? I agree with you, but this one.

What about this one? Yes, it's a pattern, isn't it? Dog, footprint, footprint, dog, footprint, footprint, dog, footprint, footprint? But if we look at it like this, is this one a pattern yet? You're right, it could be the beginning of a pattern, but because we can't see anything going over and over and over again, we don't know yet. So that's a really important thing about patterning that we see the core repeating over and over and over.

So let's explore some patterns. What I have hiding underneath here is a pattern that I've made using shapes. So it's important that you know that it's a pattern, so you're looking for the part that repeats over and over and over again, so we can already see this shape here. What's that one there? That's right, it's a square. And what I wonder is what do you predict might come next? I hear what you're thinking, so some of you are thinking it could be a circle. Some of you are thinking it could be a rectangle, some of you were thinking it could be a hexagon, and I heard some people say they thought it could be a triangle.

Let's have a look. It's a triangle! So in my pattern, it's starting off with a square and a triangle and we know that it's a pattern, so there's something inside of this has to be repeating over and over and over. Do you know what comes next for sure? That's right, you don't yet do you. 'cause all we know is it's got a square and a triangle, so we haven't seen the part yet that repeats over and over and over so we could have a pretty good prediction though. What are you thinking this time? Oh, I see. Some of you are thinking that what might come next is a square. Some of you are thinking that what might come next would be a circle. Some of you are thinking it could be a triangle. Let's see what happens. A square that's right. So square, triangle, square. What do you think could be next in my pattern? Let's see what happens. Oh, a triangle, are you starting to feel more confident about how this pattern is emerging?

Yes, OK, I see why because we now have square, triangle, square, triangle. But it's always good to check to see what comes next so that we can see something over and over and over again. So let's see. Ah, you think it's a square? Let's see what happens, oh. It is! And what do you think might come next now? A triangle? And you're feeling much more confident. Let's see. It's a triangle, so let's see. We now have square, triangle, square, triangle, square, triangle and we can see that it repeats over and over and over. So we can pretty much trust now that we've discovered what the core of our pattern is.

That's right. Is that what you were thinking too? The square and the triangle? So let's see what comes next. It has a square, oh and, hmm, I wonder what would come here if we continued our pattern? Square, triangle, square, triangle, square, triangle, square. Ah, I heard you too. A triangle. So in this sort of a pattern can you see where the core is?

Yeah, the square followed by the triangle and a way that we can check this as mathematicians is to move our shapes around. So what I like to do in my head is to or actually with my equipment is to move things around so that I can check that I can see the core, 'cause sometimes we can work with some really tricky patterns or it's hard to work out what the repeating core is. So as a mathematician I can always think flexibly about these situations. So what I might do is start by,

if I think my core is a square and a triangle, I can move down my other square in a triangle, and my next square and triangle and my next square and triangle, and what I'm hoping to see the objects here are the same and these ones here also have the same attribute. So in this case all of my shapes are squares and in this case all of my shapes are triangles and so I can see that I have a pattern. The core is a square and a triangle, and that's the part that repeats over and over and over again, and I can move it back into my line to check.

I wonder if I could use that strategy to help me work out what's missing? Alright, so mathematicians close your eyes. I can still see some of you are peeking, I'm going to cover this up. And let's see. OK, so now one of my shapes in my pattern sequence is missing and it's this one. Right there, sometimes I like to put something in its place so I know this is what's missing and what might it be. So can you work it out? Oh, I can hear you. Some of you think it's a square because you're thinking square, triangle, square, triangle, square, triangle, square, triangle. It's really nice reasoning we could use our strategy of moving our pattern core to check and to prove that it is a square that's missing. So let's do that. So we move this chunk of our core, or our pattern, our core, we move this part, including the bit that's missing and we can move this part down here. That's right, and now it's really easy for us to see isn't it? That this part here that's missing should definitely be a square, because everything has to be or have the same attributes. That's right, and so if we put this back, we can now say this one here that was missing was in fact a square.

Do you want to try that again? OK, close your eyes and just in case, I think I'm going to make this one tricky or challenging, which is really nice 'cause we love it when our brains sweat as mathematicians, OK? Oh yeah. Alright, little mathematicians. What are you thinking now? That's right, so you can see this time there's 2 objects missing aren't there? So one missing from here and the other's missing from there. That's right, and this time it can sometimes be a bit harder to work out what's missing because we don't get that repeating sort of sound of square, triangle, square, triangle, square, triangle to help us out because it's this fourth one here, that's missing. Ah yeah, but we've heard it a few times now haven't we? Should we use our strategy of moving our core around to check? Ok.

So, so let's imagine this, where here's what we think our pattern core is, the square and the triangle, well we know it is 'cause we've established that and we'll move these down to here and this down here and these ones down here and now. We can see that's right, this one here we should replace with a... triangle, and this one should be a? ...square. That's right.! And if I move these back up to here, this one here that we thought was missing was a triangle and this one that was missing, is a square. It's a really interesting strategy, isn't it? About ways of investigating things that are missing.

I wonder if we could use that on a pattern that's a little bit trickier. Let's see so I've got some other things here, and I'm going to make myself a pattern - stand by mathematicians. Can you see that pattern now? OK, and what are you thinking, first of all, is the core of our pattern? Should we read it together? OK. Square, triangle, triangle. Square, Triangle, Triangle. Square, Triangle, Triangle. Square, Triangle. So if it was a pattern like before, where we had two things in our core, I could move these like this, and already can you see that too? Yeah, what's the problem? Yeah, that's right, this should be a square. If it was still a pattern. So, so that one didn't quite work, I wonder... Ah, I hear what you're thinking. You think there might be 3 things inside the core? Let's see. So if it goes square, triangle, triangle, these things should move down here. Does that look better? I see, and these things too? Square, triangle, triangle and then... square, triangle. If I wanted to continue my pattern, what would I need next? Ha, and how do you know that? Yeah, I was thinking it's really nice thinking, little mathematicians because, that's right, you can see they're all triangles, and so that would be a triangle to continue my pattern. So let's put this back up here now. So this time my pattern core has a square, triangle, triangle in it. So three things inside it. And let's see if we can use this strategy of moving our pattern core around to help us work out what things are missing. Alright, close your eyes.

Good job. OK. Let's see, so this is the shape that's missing from our pattern now. What do you think it is? Should it be a square or a triangle? And can you explain why you think that? Oh, I could hear someone saying that they imagined moving these things down here. So if we moved this square down to there that would align. And if we, this space that's missing here would be underneath a?...triangle and this one would be underneath a triangle too, so that dot must represent a triangle? Huh, I like the way that you're visualising like a mathematician. Let's move this down and check. Should we move the other parts of our repeating core too? Yeah, so just to make sure. Huh? And so because, that's right, because the core is in alignment. That part there should be a?... Triangle. OK, let's push it back out to see. Yeah. And that works. It's really nice work today, little mathematicians.

I wonder if you could go now and make some patterns of your own, repeating patterns of your own and work out and get someone at home to help you work out and draw somewhere some of the pieces are missing or have someone make them for you where some of the parts are missing and you can go and investigate and use your mathematical thinking to help you solve how to continue a pattern and also how to work out things that are missing from a pattern sequence? Over to you, little mathematicians!