Staircase patterns

Hello there mathematicians. How are you today? That's really good to hear. So, I have some blocks here and I'm beginning to make a pattern.

Yeah, and I'm wondering if you can figure out what the next term or the next thing would be in my pattern if I continued it? Ah ha, so remember, a pattern is something that has a repeating core. So we're looking for what is the thing that's repeating over and over and over again. Yeah, and it's a bit different 'cause we've been working a lot on patterns like A,B, A,B. You know, like clap, click, clap, click, clap, click. But this is a different kind of pattern.

Ah, you think you can see something? Uh, huh. I think, I think I can see what you're seeing too. That there's one brown cube, two black cubes, three red cubes. So the colour is changing each time, so there's nothing there that's repeating. But it goes from one to two three and it's, yeah, there's one more block each time. Let's use our chunking strategy to see if that works. If I line them up. Does it increase by one more block each time? Ah, so what would be the next one? It would be three and one more. Four. Ok, I can get four together. And what would be the next term in my pattern?

Four and one more. Five. The number after four is five. Ah ha, look at that. So that's what it would look like if we put it all back together and we could keep building it out as a big tower of pattern. Oh, it reminds you of something else. It also. Yes, I agree with you. It reminds me too. If I turn it like this, you might have recently seen an episode of number blocks called Step Squad. Yes, and it does look exactly like our number blocks characters, doesn't it?

Let's work with them 'cause they're cute. So when you see our number blocks characters in a step squad formation, what are some things that you notice about the shape that they have? Ah yeah, it's like a triangle.

Let me write this down. So. How many bricks? How many steps do I need to draw? One for red, another one for orange, another one for three, another one for four and one for five, and so it's like, oh, a triangle with squares in it. So it's like a triangle shape. What else do you notice about it?

Oh yeah, it's like this. It goes up by one each time. Um, each column increases by one. Look, it's one, then two, then three. Yes, so each column increases by one. So each time we go up a step, it gets one bigger. What else can you see? Oh look, you're right. one, two, three, four, five blocks wide and one, two, three, four, five blocks high. That's cool. So five blocks wide and five blocks high. Oh, yes, and I can come in and mark those for you, so I'm going to come in and go, there and half and quarter. So, one, two, three, four, five.

One, two, three, four, five, yeah, and you're right, because it's a mathematical drawing, I don't have to draw all of the individual blocks. I can just draw the most important information. Yeah, and, and so, what our pattern is, our pattern core. How would we describe that? Yeah, the pattern core goes up by one each time. So our core is that each step up the staircase adds one more block. That's really cool mathematicians.

So now I was wondering something actually, and I was wondering what would happen if we make it so we go up by one each time, but when we get to five, what would it look like if we went down the other side? Ah. Yes, so if we had one less than five, how many would we have? How many blocks? Four because one less than five is four. There we go, look. Yes, 'cause If we had five and we take one away, that's one less than five.

And what would come next in our pattern? Ah, yes, three, because one less than four is three. Uh-huh. Ok mathematicians, over to you to finish what the rest of our staircase would look like if it goes up and down the other side. And can you draw a picture of this to to record your thinking? Over to you.

So over to you mathematicians. Draw the staircase pattern we've made, continuing it down the other side. And then we'll come back together. OK over to you.

Welcome back mathematicians. How did you go?

Ah ha, yeah, now it looks like a triangle but in a different orientation, doesn't it? Before look, the triangle shape was here and now we see the triangle shape here. Yeah, yes, and if we continued our pattern as a growing/ shrinking pattern then it would start to grow again, before it started to, whoops, I've got to make it grow and then it has to shrink again, look. That's really cool, isn't it? It's quite beautiful too.

Oops. Uh-huh, and then I would see, yeah, you're right it looks a bit like mountains, doesn't it now? Yeah, that's really cool. Alright. so my challenge for you mathematicians, while thinking about something is, this is what our pattern looks like if we just use one block wide. But what would happen if we made a staircase like this, using two blocks wide?

Yes, so, we had one two, and then we might have two twos. Not something you would dance in, well I mean, you could. Then you might have three twos, uh-huh. So mathematicians over to you now, can you continue? You can build this structure if you like, our model or draw it in your notebook. What will it look like if I continue building it up and down the other side?

Over to you mathematicians?

Ok mathematicians does your drawing look like our model here, our structure? Oh awesome. So what I'm curious about is, do you know how when we first looked at this step squad and we thought about some things that we noticed about it. What are some things that we notice about this staircase?

Some things might be the same and some things might be different. Yeah, it still makes a triangle, doesn't it?Like this guy still makes a triangle? Yeah, but it's much wider. In fact it's twice as wide. Yes, because I have instead of one for each, one brick here I have two bricks instead. So look, we can prove it's twice as wide 'cause if we put this here to measure it yes, so we got to remember that they're not all clicked in.

That's the same width and if I move it across here, if I push these in. Whoops! Yeah, you'll see it's also the same width. Uh huh. So it's twice as wide. Ah, yeah, it's the same height, isn't it? 'Cause if we take five and lay it here, it's still only five bricks high, but if it's twice as wide, how wide is it?

Ah, eighteen, yeah, 'cause look this is five, six, seven, eight, nine and double nine is eighteen. Would you like to count to check? Ok, ready? Two, four, six, eight, ten, twelve, fourteen, sixteen, eighteen bricks wide, blocks wide. Ah ha.

What else do you notice? Yeah, Ok, well I was noticing, I want to move these guys out of the way. Thinking about something else that I thought was really interesting mathematicians, I'm just going to move these across a little bit. Is, I was noticing something about, depending on how we look at the structure. So when we look at it this way it goes up and then it goes down. But if we look at it this way, I can see that this row is longer than this row which is longer than this row which is, longer than this row, which is longer than this row.

Uh-huh, look. And if I squish it up like this way that makes it easier to see. Ah, yes, and now it just looks like each time I go up it shrinks each time or each time I go down it grows. Ah ha, so, I can see it as either a growing or a shrinking pattern.

Ah ha, so let's work out how many blocks there are in each row. What do we know about the bottom? Yes, we know that's eighteen, we just worked out that it's eighteen blocks wide. And what do we know then about this row here? Yeah, we're not going to count just yet 'cause we know something, don't we?

We know, let me use these two bricks, that if I had these two blocks here. I'll just move these out of the way for a moment. This row would be as long as this row wouldn't it? Which would mean it's eighteen but I need to get rid of two, sixteen, and remove another two. Ah ha, that makes fourteen. Yes, 'cause we're counting backwards in twos. You could have done that strategy. Ok, so eighteen and fourteen.

And then what about this next row? What do we know about it? Yes, we know that if we had two more twos, it would be the same length as this row. But there's one two missing and another two missing, which is four. Yeah, and you could use place value knowledge this time because one ten and four is called fourteen.

So just, if you get rid of the four, you just have one ten left, which is ten. Ok, so know that row has ten. And what do we know about this one now? Uh-huh it would have ten if it had two more twos. But it has two twos missing, which is four, so it's, six. And what about this guy? Yes, yeah, I know you can see how many there are. There's two.

So, I have a challenge for you mathematicians, to think about. What I'm wondering about now, is how could we work out how many blocks there are altogether? Yes, by using some efficient strategies to help us reason our way through. Ok, Mathematicians over to you to have a think and record your ideas, and then we'll come back together. Ok.