Staircase patterns

Alright mathematicians, welcome back. Let's talk about some strategies you could have used to solve that problem. So as mathematicians, we know we always like to think about what do I already know and how can I use that to help me? Ah ha. So, when you're looking at these numbers, you might have noticed something like eighteen and two more, combines to make twenty. So what we could do is have moved this block down here and now we have one row of twenty.

Yes, and we could have done that in another place as well, look. Yeah, look, six and fourteen combines to make twenty. Yes, 'cause we know six and four makes ten and then one ten more is twenty. So, I could imagine my fourteen and my six. There's the fourteen and there's the six. Yes, and then we have the ten left. Look, uh-huh. So now I know I have twenty, forty and ten more is, fifty. Uh huh, so that's one way we could have worked it out.

I'm going to put these back where they belong. The colour's nice to help me. So one way was we could look for twenty. So, so let's let's write that down together. So one strategy was to say, well I know eighteen and two is equivalent to twenty. And I know fourteen and six more is equivalent to twenty and then I have ten left and then I know that two tens, plus two tens, plus one ten, is five tens and that's fifty.

So that's one strategy we could have used. I wonder if there's another one? Ah yes, some of you guys did some imagining. I thought about this strategy too. I'll just put these back here. Yes, where you imagine things moving. Yes, like if we took the four, twos, and move them, yes, you are required, yes to make those sounds, over to here. I now have one, two, and four, twos, which is five twos. Ah ha, and it looks like, yes, a ten-frame. What would you do now to the others? Uh-huh, yes, if we take the three twos and join it with the two, twos that makes five twos, which is another ten. And then? Yes, these two twos, that always makes me think about ballerinas, come and join these three twos to make another five twos. And then? Yes, this one two joins the four twos and then there's five twos left. So, one ten, two tens, three tens, four tens and five tens, which is fifty.

So what we thought about here was making groups of tens, didn't we? So we, we first joined, two and eight. So we had one two plus four twos to make ten or one ten. Then we had, two twos plus three twos, to be one ten.

Then we have, three twos plus two twos to be one ten. And then, uh, huh. Four twos and, and one two to be one ten and then five twos which is one ten. And then we have one ten, two tens, three tens, four tens, five tens is fifty.

Wow. They're two really great strategies for thinking about our problem and you know, mathematicians. This one made me think about something else, that now, look how our step squad turned into a rectangle. And if I get at our original step squad. Here they are. Does the same thing happen? Does it turn into a rectangle? Shall we try? Ok, we'll match the one with the four because it's five bricks high. And then three and two. Uh-huh, two and three. Wow, look, it still made a rectangle, but it's a special kind of rectangle called a square. Yeah, look 'cause it's five wide and five tall. Whereas here it's ten wide and five tall.

Ah, and mathematicians this is making me think about our next challenge.

So what's some of the mathematics we just saw? Yeah, we realise over these few sessions that you can have growing patterns. We saw a growing pattern when we looked at a staircase structure like this. We can see that each time you go up one step, the number in each column or tower increases by two. Yes, and we can also see shrinking patterns. Yes, and that's when each tower decreased by two. So shrinking patterns occur when the value in each term decreases by something.

And we also realise that you can sometimes describe something as a growing or a shrinking pattern. With this up and down staircase, I can read, this is each row increases by two, increases by four each time if I start at the top and read down. So there's two, six, ten, fourteen and eighteen. I could also read this same structure from the bottom up, meaning I'd see a shrinking pattern because each row decreases by four, if I start from the bottom.

We also realised today that mathematicians use what they know to help them solve problems. So since I know twenty can be composed of eighteen and two more, I could use that knowledge. I also know that twenty can be composed of fourteen and six more, so I could use that as well. I also know, that two tens and two tens and one ten is five tens. I know that that's renamed as fifty. I could also think about this problem differently using other knowledge I know. I know one two and four twos is five twos which is a ten. I know two twos and three twos is five twos which is another ten. I also know three twos and two twos is five twos which is a third ten. I know four twos and one two is five twos which is a fourth ten and I know five twos is the same as saying one ten. Yes, and then all I had to do was rename it as fifty 'cause I know five tens is called fifty.

So mathematicians, your first challenge is to use objects to create a new staircase structure. Draw it and describe the things you notice in your workbook. Then think about how you can describe your pattern or the staircase structure that you've made, both as a shrinking pattern, and as a repeat and as a growing pattern.

Over to you mathematicians.