About how many paperclips?

Hello little mathematicians, welcome back.

I hope you're having a really lovely day. I've got this piece of paper here for us to do some thinking with and I found these paper clips at home and what I was wondering or thinking about is that we could use our paper clips and this piece of paper to warm up our mathematical imaginations and so we know when read stories, it's really important to imagine what's happening in stories in our brains 'cause it helps us make meaning from them. And the same happens in mathematics, it's really important for us to work on and practice our mathematical imaginations. So I wonder if you can help me by thinking about, about how many paperclips long is this edge of my piece of paper?

What are you thinking? Ah okay, well let's test out your theories. So it's one paper clip and two paper clips, and yes, I was being very careful not to overlap them or leave big spaces when I'm measuring. You've got to be very accurate. Two paperclips, what are you thinking now?

Okay, I like how you're using evidence to refine your estimates. Three paperclips. Four paper clips. Fa...

Ah, you're right mathematicians. Good spotting. I can't measure the length of my piece of paper using different sized paper clips. So you're right. What we should have done is sorted them out first. So I'm looking for ones that are the same length. These ones are not the same length as those big ones. They are definitely shorter. They're all the same length about. I think this one is like the big ones. Can you see any other big ones? Yeah, thank you. This one.

It's good sorting task, isn't it? I'm sorting them based on the their, their attribute of length, so that's why some of them in here you can see ones that are yellow and blue and pink because we're not worried about the colour, we're worried about how long they are. That's really important when we're measuring length.

Green one. This one, this one, I think this ones a long one. Oh,oh my gosh is that the same length? No, it's a bit shorter than the long ones and it's... oh, it's about the same length, isn't it? It's a little bit bigger than these ones here, so I'd love to be able to use my plane paper clip, but I'll put that away for now. So you're right, what I should be using is all the same size, so let's go back to using these ones 'cause that's where we started. So we had worked out I think, that we had three of them.

And now how many more do you think I need to measure the length of my line? Yes, some of you are thinking about three more 'cause that looks like that has about the space to fit three more. And some of you think it's four 'cause it's a little bit bigger than three. You think? OK, let's work it out to check.

What are you thinking now? Now you think just two 'cause now when you see that you think I can't fit three more paper clips that length in there, let's see. Nice visualising mathematicians. Six paperclips long, but you know, I wonder now because I have these shorter ones about how many shorter paperclips do you think I would need to go the length of my paper?

Can you pick up your pencil and paper now and draw a picture and show me about how many small paperclips will it take to go the length or the distance of the edge of my paper. Over to you mathematicians.

Welcome back mathematicians.

Can you please show me your drawings?

Thank you, Okay, well, let's gather some more evidence now and check and see if you would refine your thinking any further.

And you know I notice something while you were busy drawing, I notice something going on just over here in this section. I'll cover this part so we know to look here and look. There's two big paper clips and they are the same length as three smaller paper clips.

Yeah, and so what I started to wonder is that does that mean that I would have another 1, 2, 3 paperclips along this distance and another 1, 2, 3 paperclips along this distance?

And then I started using my mathematical imagination to say, well, if there's three here, then there'd be another three and another three, and I know three and three combines to make six because it's like patterns on a dice. And there's 2 threes and that's six and then three more 6, 7, 8, 9, so I think I could use my mathematical imagination and this idea that for two big paperclips there were three small ones to help me reason.

Shall we check now together? That would be great. Okay, so another three.

That's right, I'm gonna make sure line them up nicely where they're touching, no big gaps, no overlaps. Oops they're a bit fiddly? And three more. And look at that.

Nine small post, paper clips 1, 2, 3, 4, 5, 6, 7, 8, 9 and six big paperclips, 1, 2, 3, 4, 5, 6. And for each of my two big paperclips, there's three smaller ones.

What an amazing mathematical discovery and good warming up of your mathematical imaginations.

Let's get ready for next challenge. And before you leave me, whoops, some of the mathematics.

Yeah, we noticed some interesting things. You can measure the length of the edge of the paper using different size paper clips. The paper is either six big paperclips long or nine small paperclips long. And really important, we notice this relationship that for each group of two big paper paperclips, there were three smaller paper clips. Really cool noticing today. Nice work mathematicians.